# Learning a Common Dictionary for CSI Feedback in FDD Massive MU-MIMO-OFDM
Systems

Pavan Kumar Gadamsetty, Student Member, IEEE and K. V. S. Hari, Fellow, IEEE
Lajos Hanzo, Life Fellow, IEEE

###### Abstract

In a transmit preprocessing aided frequency division duplex (FDD) massive
multi-user (MU) multiple-input multiple-output (MIMO) scheme assisted
orthogonal frequency-division multiplexing (OFDM) system, it is required to
feed back the frequency domain channel transfer function (FDCHTF) of each
subcarrier at the user equipment (UE) to the base station (BS). The amount of
channel state information (CSI)111We use the terms CSI and FDCHTF
interchangeably to be fed back to the BS increases linearly with the number of
antennas and subcarriers, which may become excessive. Hence we propose a novel
CSI feedback compression algorithm based on compressive sensing (CS) by
designing a common dictionary (CD) to reduce the CSI feedback of existing
algorithms. Most of the prior work on CSI feedback compression considered
single-UE systems. Explicitly, we propose a common dictionary learning (CDL)
framework for practical frequency-selective channels and design a CD suitable
for both single-UE and multi-UE systems. A set of two methods is proposed.
Specifically, the first one is the CDL-K singular value decomposition (KSVD)
method, which uses the K-SVD algorithm. The second one is the CDL-orthogonal
Procrustes (OP) method, which relies on solving the orthogonal Procrustes
problem. The CD conceived for exploiting the spatial correlation of channels
across all the subcarriers and UEs compresses the CSI at each UE, and upon
reception reconstructs it at the BS. Our simulation results show that the
proposed dictionary’s estimated channel vectors have lower normalized mean-
squared error (NMSE) than the traditional fixed Discrete Fourier Transform
(DFT) based dictionary. The CSI feedback is reduced by 50%, and the memory
reduction at both the UE and BS starts from 50% and increases with the number
of subcarriers.

###### Index Terms:

Wideband, frequency domain channel transfer function (FDCHTF), channel state
information (CSI), compressive sensing (CS), massive MIMO, common dictionary
learning (CDL), common dictionary (CD), orthogonal Procrustes (OP) problem,
K-SVD algorithm.

##  I Introduction

Massive multiple-input multiple-output (MIMO) systems constitute a promising
enabling technique for 5G/6G cellular networks as a benefit of their
substantial spatial multiplexing gain [1] in both time division duplex (TDD)
and frequency division duplex (FDD) scenarios. At the base station (BS),
combining the massive MIMO technology with orthogonal frequency-division
multiplexing (OFDM) is capable of transmitting multiple data symbols to
multiple UEs on the same time-frequency resource block, resulting in increased
system throughput [2]. In a multi-user (MU) massive MIMO-OFDM system, the
knowledge of CSI is needed at the BS to implement transmit precoding (TPC) for
suppressing the co-channel interference (CCI) [3]. In FDD systems, due to the
absence of channel reciprocity [4], the user equipment (UE) has to feed back
the downlink (DL) frequency domain channel transfer function (FDCHTF) of each
subcarrier to the BS. Feeding back the accurate CSI becomes more challenging
with the increased number of antennas, subcarriers, and UEs [5] .

The compression of high-dimensional CSI is essential for reducing the CSI
feedback. The wireless channels can be represented in a sparse form in the
spatial-frequency domain using ‘sparsifying’ bases termed as a dictionary [6].
In the compressive sensing (CS)-based feedback schemes [7], the original CSI
is mapped to a sparse domain using a dictionary. The traditional choice of the
dictionary is a fixed Discrete Fourier Transform (DFT) matrix. Then a random
Gaussian measurement matrix is introduced to compress the sparse vector for
feeding it back to the BS at a reduced rate. The sparse signal is then
reconstructed at the BS using CS-based algorithms, such as the orthogonal
matching pursuit (OMP) [8], basis pursuit (BP) [9] or covariance-assisted
matching pursuit (CAMP) [10] procedures. The original CSI is then
reconstructed by mapping the regenerated sparse signal back to the same
dictionary used at the UE side.

The authors of [11] proposed a rotated version of the DFT basis to provide
improved sparsity that results in reduced CSI mean-squared error (MSE) for a
narrowband multi-user system with each UE having a single receive antenna.
However, the proposed rotated basis does not exploit the antennas’ spatial
correlation. The massive MIMO-OFDM channel between a multi-antenna UE and the
BS can be represented by a matrix [12]. In such systems, the UE has to feed
back the FDCHTF of each subcarrier, which results in huge feedback overhead.
The FDCHTF feedback algorithm of a massive MIMO-OFDM single-UE system based on
multidimensional compressive sensing theory using Tucker’s tensor
decomposition model is developed in [13]. Briefly, Tucker’s tensor
decomposition exploits the structure hidden in all the dimensions of the
channel matrix and compresses it simultaneously in each dimension. The
proposed scheme has a significant feedback reduction and hence improves the
spectral efficiency. However, both the basis and measurement matrices should
be learned. The authors of [14] introduced a recursive least squares
dictionary learning algorithm (RLS-DLA) for CSI feedback. The proposed scheme
achieves a substantial reduction in feedback requirements, however it requires
the computation of large matrix inverses during the dictionary learning
process.

TABLE I:  COMPARING OUR CONTRIBUTION TO THE EXISTING LITERATURE

| [11] | [14] | [15] | [16] | [17] | [18] | Our
---|---|---|---|---|---|---|---
Massive MIMO architecture | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Spatially correlated channels | ✓ | ✓ | ✓ | ✓ |  |  | ✓
Sparsity | ✓ | ✓ | ✓ |  | ✓ | ✓ | ✓
Dictionary learning |  | ✓ | ✓ |  | ✓ | ✓ | ✓
Feedback savings | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Memory savings at UE |  |  |  |  | ✓ |  | ✓
Multi-user MIMO OFDM |  |  |  |  | ✓ |  | ✓
UE mobility |  |  |  |  |  |  | ✓

Another line of work focused on designing non-dictionary-based methods for
FDCHTF feedback [19]. In [20], an antenna grouping-based method was proposed
for reducing the feedback overhead by grouping multiple correlated antenna
elements into a single representative value. By considering a ray-based
channel model, the authors of [21] and [22] designed an angle-of-departure
(AoD) based adaptive subspace codebook for feedback compression. In [16] and
[23] the authors exploited the low-rank characteristics of a large channel
matrix for recovering the CSI at the BS.

Recent solutions include Deep Learning (DL) techniques conceived for CSI
compression and recovery using so-called the Bi-LSTM [24], CsiNet-LSTM [25],
DNNet [26], CS-ReNet [27], and DCRNet [28] frameworks222For the expansion of
these acronyms please refer to the relevant papers. Additionally, the
application of Deep unfolding techniques has also shown promising results, as
demonstrated in [29, 30]. These techniques have better reconstruction
performance than the conventional CS algorithms of [31], albeit at
significantly increased computational complexity.

Massive MIMO-OFDM channels tend to be individually sparse and simultaneously
share a common support set that typically exhibit joint sparsity in the time
domain (TD) [32], which results in correlation among subcarriers in the
frequency domain (FD). Since the DFT dictionary does not exploit spatial
correlation across antenna arrays, we design a dictionary for massive MIMO-
OFDM systems that can exploit the spatial correlation, hence achieving
improved CSI reconstruction performance. The dictionary is generally learned
from a training data set by relying on learning-based approaches [33, 34, 35,
17, 18]. The dictionaries learned have the potential to offer improved
normalized mean squared error (NMSE) performance compared to fixed
dictionaries, like the DFT-based one. In [15] a CS-based method was proposed,
which exploited the spatial correlation among the antennas in a narrowband
single-UE system using the K-SVD [36] algorithm. The method relies on learning
the K-SVD dictionary from the training data set and on feeding back the K-SVD
dictionary learned at the UE to the BS. Using this K-SVD dictionary, the CSI
is compressed at the UE and reconstructed at the BS. The motivation for this
K-SVD based dictionary is not only to reduce the CSI feedback, but also to
reduce the NMSE of CSI reconstruction.

TABLE II:  LIST OF ACRONYMS Acronyms | Meaning
---|---
AOD | Angle-of-Departure
BER | Bit Error Rate
BS | Base Station
CD | Common Dictionary
CDL | Common Dictionary Learning
CFR | Channel Frequency Response
CIR | Channel Impulse Response
CS | Compressive Sensing
CSI | Channel State Information
DFT | Discrete Fourier Transform
DL | Deep Learning
FD | Frequency Domain
FDD | Frequency Division Duplex
FDCHTF | Frequency Domain Channel
| Transfer Function
GR-SVD | Golub-Reinsch SVD
K-SVD | K Singular Value Decomposition
MIMO | Multiple-Input Multiple-Output
MP | Matching Pursuit
MSE | Mean Squared Error
MU | Multi User
NMSE | Normalized Mean Squared Error
OF | Objective Function
OFDM | Orthogonal Frequency Division
| Multiplexing
OMP | Orthogonal Matching Pursuit
OP | Orthogonal Procrustes
QuaDRiGa | Quasi Deterministic Radio
| Channel Generator
RA | Receive Antenna
SU | Single User
TA | Transmit Antenna
TD | Time Domain
TDD | Time Division Duplex
TPC | Transmit Precoding
UE | User Equipment
VQC | Vector Quantization Codebook

As the channel-induced dispersion is increased, the number of OFDM subcarriers
also has to be increased to avoid an excessive performance degradation. Hence
upon using the K-SVD algorithm of [15], the number of subcarrier K-SVD based
dictionaries increases as the number of subcarriers increases. Handling
ubiquitous subcarrier K-SVD based dictionaries is cumbersome in terms of
memory management and feedback load. To circumvent this problem, we propose a
novel common dictionary learning (CDL) technique, which can replace the
requirement of individual subcarrier K-SVD dictionaries, leading to the
concept of a common dictionary (CD). The CD effectively captures the channel
characteristics of all the subcarriers and UEs, making it the optimal
sparsifying dictionary for representing the channel’s sparsity in massive MIMO
systems. Given the CD learned, compressive channel estimation techniques can
be constructed for acquiring the CSI. A set of two methods having different
pros and cons are proposed for CDL, namely the CDL-KSVD method and the CDL-
orthogonal Procrustes (OP) [37] based method. These methods are detailed in
Section III of the paper. Again, our primary motivation is to reduce the CSI
feedback overhead on the uplink as well as the memory requirement at both the
UE and the BS in FDD massive MU-MIMO-OFDM systems.

TABLE III: LIST OF NOTATIONS $N_{t}(N_{r})$ | Number of TAs (RAs)
---|---
$L$ | Number of channel taps in TD
$i$ | Index of the channel tap
$N_{c}$ | Number of subcarriers in FD
$l$ | Index of the subcarrier
$K$ | Number of UEs
$k$ | Index of the UE
$V$ | Velocity of the UE
$S$ | Sparsity index
$n$ | Frame index
$\bm{\Psi}$ | Dictionary
$\bm{\Phi}$ | Measurement (Sensing) matrix
$\bm{\Theta}$ | Product of a dictionary and $\bm{\Phi}$
$N_{g}$ | Number of rows in $\bm{\Phi}$
$g$ | Compression factor
${M}^{\prime}$ | Total number of training channel vectors
$N$ | Number of training channel vectors
| considered across each subcarrier
UEk |  $k$-th user equipment
$\mathbf{R}_{BS}$ | Spatial correlation matrix at the BS
$\mathbf{R}_{UE,k}$ | Spatial correlation matrix at the $k$-th UE
$\mathbf{\bar{H}}_{n,k}$ | TD channel matrix across the $n$-th tap
| of the $k$-th UE
$\mathbf{\breve{H}}_{n,k}$ | TD i.i.d channel matrix
${\mathbf{H}}$ | Concatenated channel matrix of $K$ UEs
$\mathbf{H}_{k}$ | Channel matrix of the $k$-th UE
$\hat{\mathbf{H}}_{k}$ | Reconstructed channel matrix of the $k$-th UE
| at the BS
$\mathbf{H}^{\prime}$ | Matrix of training channel vectors at the UE
$\tilde{\mathbf{H}}^{\prime}$ | Sparse channel matrix representation of $\mathbf{H}^{\prime}$
$\hat{\mathbf{{H}}}^{\prime}$ | Matrix of reconstructed training channel
| vectors at the BS
$\hat{\tilde{\mathbf{H}}}^{\prime}$ | Sparse channel matrix representation of $\hat{\mathbf{{H}}}^{\prime}$
$\mathbf{H}_{l_{n},k}$ | Channel matrix or FDCHTF across the
|  $l$-th subcarrier of the $k$-th UE in $n$-th frame
$\hat{\mathbf{H}}_{l_{n},k}$ | Reconstructed channel matrix across the
|  $l$-th subcarrier of the $k$-th UE in $n$-th frame
${\mathbf{h}}_{l_{n},k}$ | Channel vector transformation of ${\mathbf{H}}_{l_{n},k}$
$\tilde{\mathbf{h}}_{l_{n},k}$ | Sparse channel vector representation of ${\mathbf{h}}_{l_{n},k}$
$\hat{\mathbf{h}}_{l_{n},k}$ | Channel vector transformation of $\hat{\mathbf{H}}_{l_{n},k}$
$\hat{\tilde{\mathbf{h}}}_{l_{n},k}$ | Sparse channel vector representation of $\hat{\mathbf{h}}_{l_{n},k}$
$\mathbf{h}_{c,l_{n}}^{k}$ | Compressed channel vector across the
|  $l$-th subcarrier of the $k$-th UE in $n$-th frame
$\bm{\Psi}_{DFT}$ | DFT dictionary
$\bm{\Psi}_{ksvd,l}^{k}$ | K-SVD dictionary across the $l$-th subcarrier
| of the $k$-th UE
$\bm{\Psi}_{c}$ | Common dictionary
$\Delta_{saved}$ | Total memory storage reduction
$\Upsilon$ | Dictionary feedback reduction factor
$\mathcal{T}_{ksvd}$ | Feedback required for sending
|  $N_{c}$ subcarrier dictionaries
$\mathcal{T}_{com}$ | Feedback required for sending $\bm{\Psi}_{c}$
$\mathcal{T}_{saved}$ | Total dictionary feedback reduction

Main contributions of this article:

  1. 1.

We proposed a novel CDL framework for learning a CD, mainly using the CDL-KSVD
and the CDL-OP methods. In the CDL framework proposed for a multi-UE system,
the CD conceived exploits the spatial correlation of the FD channels across
all the subcarriers and the UEs. We demonstrate that this implementation
improves the NMSE performance when compared to the existing methods.

  2. 2.

In the CDL framework proposed for a single-UE system, the learning of CD is
implemented at the UE. The UE sends only the CD to the BS in the uplink
instead of all the subcarrier K-SVD dictionaries. This implementation reduces
the dictionary feedback to the BS by a factor of $N_{c}$ and also reduces the
memory requirement by having a single CD at the UE and the BS.

  3. 3.

We evaluate the proposed CD in the context of various system configurations
and channel conditions in the face of UE mobility. The numerical results show
a significant reduction in the NMSE of channel estimation and highlight the
bit error rate (BER) performance of the channel estimates when using our
learned dictionary. This corroborates the effectiveness of the CDL framework
proposed over existing methods in wideband massive MIMO systems.

The remainder of this paper is organized as follows. Section II presents the
system model, CS procedure, and the motivation. In Section III, the proposed
methods are discussed. Then the application of the proposed methods in
wideband systems is discussed in Section IV. Our simulation results are
provided in Section V to show the NMSE performance of the proposed method
compared to state-of-the-art methods. Finally, in Section VI, our conclusions
are given.

Notations: We use lower (upper) bold letters to denote column vectors
(matrices) and super-scripts ${(.)^{-1}},{(.)^{*}},{(.)^{H}}$ to represent the
inverse, complex conjugate and Hermitian operators respectively, $\|.\|_{F}$
denotes the Frobenius norm of a matrix; $\otimes$ denotes the Kronecker
product, tr(.) is the trace of the matrix, vec(.) operation returns a column
vector by stacking all the columns of a matrix.

Figure 1: Overview of the considered massive MU-MIMO-OFDM system. Right: $K$
UEs with $N_{r}$ RAs each for $k$ = $1$ to $K$; Left: massive MIMO base
station with $N_{t}$ TAs. $\mathbf{H}_{k}$ represents the complete FDCHTF of
the $k$-th UE and $\mathbf{H}_{l,k}$ represents the FDCHTF across the $l$-th
subcarrier of the $k$-th UE.  TABLE IV:  References for CSI compression
techniques

| Single-UE | Multi-UE
---|---|---
Narrowband |  | CS basis [14], [31]
---
| Rotated DFT [11]
---
K-SVD dictionary [15]
Bi-LSTM [24]
Wideband |  | Multidimensional CSI [17]
---
CsiNet [38]
CsiNet-LSTM [25]
DNNet [26]
CS-ReNet [27]

##  II System Model

In this section, we first introduce the massive MU-MIMO-OFDM channel and the
associated spatial correlation matrices at the BS and UEs. Furthermore, we
conceive the CS-based channel reconstruction procedure of massive MIMO
channels. Next, we highlight the dictionary learning algorithms available in
the literature. Then, in the final sub-section we describe the motivation of
the proposed CDL framework.

###  II-A The Massive MU-MIMO-OFDM Channel

We consider a massive MU-MIMO-OFDM system using a uniform linear array (ULA)
of $N_{t}$ TAs at the BS, $N_{r}$ RAs at all the $K$ UEs and $N_{c}$
subcarriers. For the $k$-th UE ($k$ = $1$ to $K$), consider a frequency-
selective channel having $L$ taps in the TD. Let $\mathbf{H}_{l,k}$ represent
the FDCHTF of the $l$-th subcarrier of the $k$-th UE given by

$\mathbf{H}_{l,k}=\sum\limits_{i=0}^{L-1}\mathbf{\bar{H}}_{i,k}e^{-{j2\pi
il\over N_{c}}},$ (1)

where $\mathbf{\bar{H}}_{i,k}\in\mathbb{C}^{N_{r}\times N_{t}}$ is the $i$-th
tap TD channel matrix. The tap coefficient $\mathbf{\bar{H}}_{i,k}(p,q)$
represents the channel impulse response (CIR) of the link spanning from the
$q$-th BS antenna to the $p$-th UE antenna.

The spatial correlation of massive MIMO channels can be modeled by a Kronecker
structure having separable transmit and receive correlation matrices [31],
with $\mathbf{\bar{H}}_{n,k}$ given by

$\mathbf{\bar{H}}_{i,k}=\frac{1}{\sqrt{\textrm{tr}(\mathbf{R}_{UE,k}})}\mathbf{R}_{UE,k}^{\frac{1}{2}}\mathbf{\breve{H}}_{i,k}\mathbf{R}_{BS}^{\frac{1}{2}},$
(2)

where $\mathbf{\breve{H}}_{i,k}$ is a $N_{r}\times N_{t}$ matrix whose
elements are independent and identically distributed (i.i.d.) complex zero-
mean, unit variance, Gaussian random entries. Furthermore, $\mathbf{R}_{BS}$
and $\mathbf{R}_{UE,k}$ are the spatial correlation matrices at the BS and
$k$-th UE, respectively.

The spatial correlation matrices are generated by Jakes’ model often used in
the literature, so the $uv$-th element of $\mathbf{R}_{BS}$ and
$\mathbf{R}_{UE,k}$, can be modeled by $r_{uv}=J_{0}(\,{2\pi
d_{uv}}/{\lambda})$ where $d_{uv}$ is the distance between the antennas $u$
and $v$, $\lambda$ is the carrier wavelength and $J_{0}$(.) denotes the
zeroth-order Bessel function of first kind [39].

Figure 2: Overview of the considered CSI feedback compression scheme in the
massive MU-MIMO-OFDM system. The FDCHTF across the $l$-th subcarrier of the
$k$-th UE is compressed at the UE and reconstructed at the BS.

Figure 3: Overview of the massive MU-MIMO-OFDM system having a $k$-th UE with
$N_{r}$ RAs and a BS with $N_{t}$ TAs. $\mathbf{H}_{k}$ represents the
complete FDCHTF at the $k$-th UE and $\mathbf{H}_{l,k}$ represents the FDCHTF
across the $l$-th subcarrier of the $k$-th UE. The FDCHTFs are vectorized and
then undergo sparse transformation using the dictionary obtained from the
existing methods 1, 2 of [15] and the proposed framework. The circled numbers
and the notations are the same as in Fig.2.

###  II-B Compressive Sensing Based Channel Reconstruction

Fig. 2 shows the basic schematic of the FDCHTF compression and reconstruction
across the $l$-th subcarrier of the $k$-th UE. More specifically, observe in
Fig. 3 that at the $k$-th UE we present the FDCHTF view across all the
subcarriers and its sparsification using the existing as well as the proposed
methods in parallel. The remainder of this section introduces each of the
steps numbered in both figures.

1

In between the BS and a $k$-th UE the complete channel frequency response
matrix that includes all the $N_{c}$ subcarrier channels is formed by stacking
the channel matrices column-wise:

$\mathbf{H}_{k}=[\mathbf{H}_{1,k},\;\ldots,\;\mathbf{H}_{l,k},\;\ldots,\;\mathbf{H}_{N_{c},k}].$
(3)

2

We assume that the $k$-th UE perfectly estimates its channel matrix
$\mathbf{H}_{k}$, which should be shared with the BS through feedback. Instead
of sending the FDCHTF of each subcarrier, the matrix $\mathbf{H}_{l,k}$ is
vectorized first into an $N_{r}N_{t}\times 1$ column vector using the vec(.)
operation

$\mathbf{h}_{l,k}=\textrm{vec}(\mathbf{H}_{l,k}).$ (4)

34

In practical systems, the UE has to compress the estimated channel vector
$\mathbf{h}_{l,k}\in\mathbb{C}^{N_{r}N_{t}\times 1}$ to avoid high feedback
load. The wireless channel vector $\mathbf{h}_{l,k}$ can be represented by a
sparse vector [31] after a transformation

$\mathbf{h}_{l,k}={\bm{\Psi}}\tilde{\mathbf{h}}_{l,k},$ (5)

where $\tilde{\mathbf{h}}_{l,k}$ is the sparse representation of
$\mathbf{h}_{l,k}$. The number of non-zero components of a sparse channel
vector is called the sparsity or sparsity index, and it is denoted by ${S}$,
while ${\bm{\Psi}}$ is a $N_{r}N_{t}\times N_{r}N_{t}$ dictionary known to
both the UE and the BS. A popular example of $\mathbf{\Psi}$ is the DFT
matrix. Next, we introduce the measurement (sensing) matrix ${\bm{\Phi}}$,
which plays a crucial role in compressive sensing. The measurement (sensing)
matrix defines the measurement process in CS, which influences the
reconstruction quality and efficiency of the signal recovery algorithm. It is
responsible for mapping the original high-dimensional signal to a lower-
dimensional signal.

56

To compress the channel vector $\mathbf{h}_{l,k}$, a measurement matrix
${\bm{\Phi}}\in\mathbb{C}^{N_{g}\times N_{r}N_{t}}$
$(N_{g}$$<\\!<$$N_{r}N_{t})$ that satisfies the Restricted Isometry Property
(RIP) [8], which facilitates sparse vector recovery relying on:

$\mathbf{h}^{k}_{c,l}={\bm{\Phi}}{\bm{\Psi}}\tilde{\mathbf{h}}_{l,k},$ (6)

where $\mathbf{h}^{k}_{c,l}$ is the compressed channel vector with dimension
$N_{g}\times 1$. Let us now define $\bm{\Theta}=\bm{\Phi}\bm{\Psi}$.

7

Then the reconstruction of $\mathbf{h}_{l,k}$ can be formulated as an
$\ell_{0}$-norm minimization problem and the sparse vector
$\tilde{\mathbf{h}}_{l,k}$ can be obtained by solving

$\min\limits_{\tilde{\mathbf{h}}_{l,k}}\|{\tilde{\mathbf{h}}_{l,k}}\|_{{0}}\quad
s.t.~{}~{}{{\mathbf{h}^{k}_{c,l}}}={\bm{\Theta}}{\tilde{\mathbf{h}}_{l,k}}.$
(7)

Thus, instead of feeding back $\mathbf{h}_{l,k}$, the UE sends a low-
dimensional vector $\mathbf{h}^{k}_{c,l}$ to the BS for reducing the FDCHTF
feedback. The BS reconstructs $\hat{\mathbf{h}}_{l,k}$ from
$\mathbf{h}^{k}_{c,l}$, where $\hat{\mathbf{h}}_{l,k}$ represents the
reconstructed $\mathbf{h}_{l,k}$. The reconstructed channel vector
$\hat{\mathbf{h}}_{l,k}$ at the BS is utilized for precoding during the data
transmission stage. The precoder matrices employed at the BS are denoted as
$\mathbf{W}^{g}$ and $\mathbf{W}^{g}_{op}$, which correspond to the
beamforming weights. These weights are obtained from the true channels and the
channels estimated using CDL-OP dictionary, respectively, for a compression
factor of $g$. Here, the compression factor $g$ is defined as
$g=\frac{N_{r}N_{t}}{N_{g}}$, where $N_{g}\times 1$ represents the dimension
of the compressed channel vector $\mathbf{h}^{k}_{c,l}$.

###  II-C Motivation for the Common Dictionary Learning Framework

In CS-based feedback schemes, the traditional choice of the dictionary is a
fixed DFT matrix, which does not exploit the spatial correlation between the
antennas. The authors of [11] proposed a rotated version of the DFT dictionary
for better exploiting the sparsity, resulting in reduced FDCHTF mean-squared
error (MSE) for a narrowband multi-user system supporting single antenna UEs.
But this rotated basis still failed to exploit the antenna’s spatial
correlation for improving the MSE further.

The authors of [15] have shown that a dictionary can be learned using the
K-SVD algorithm for narrowband FDD massive SU-MIMO systems. This K-SVD
dictionary learned exploits the spatial correlation between the antennas, and
its FDCHTF reconstruction performance is improved compared to the fixed DFT
dictionary. The proposed method requires FDCHTF and dictionary feedback to the
BS. However, in practical communication systems, the channels are frequency-
selective, and OFDM is a ubiquitous technique for such systems. In a massive
MU-MIMO-OFDM system, to extend the idea of dictionary learning, it is
necessary to feed back the FDCHTF and K-SVD based dictionary of each
subcarrier of all UEs. Feeding back the entire FDCHTF $\mathbf{H}_{k}$ of the
$k$-th subcarrier will be a huge burden in the uplink. Another important issue
is that substantial memory is required for saving all the $N_{c}$ subcarrier
dictionaries at both the UE and the BS. The dimension of each subcarrier
dictionary is $N_{r}N_{t}\times N_{r}N_{t}$, hence the memory required to
store $N_{c}$ dictionaries is $N_{c}(N_{r}N_{t})^{2}$.

To overcome these challenges, we propose the novel idea of a common
dictionary, which can replace the requirement of individual subcarrier
dictionaries. The CD is designed for exploiting the spatial correlation across
all the subcarriers and UEs in the FD, hence improving the CSI reconstruction
accuracy. The proposed CD reduces the CSI feedback load and memory requirement
in both single and multi-UE systems. In particular, the feedback load is
further reduced for a single-UE system by sending only a single CD from the UE
to the BS. Hence, the proposed CDL framework reduces the CSI feedback and
memory requirements, and we also study the NMSE performance compared to the
DFT and subcarrier K-SVD dictionaries.

Figure 4:  Flow of the mathematical analysis.

##  III Proposed Common Dictionary Learning Framework

In this section, we detail the CDL framework proposed for a multi-UE system
that constructs a dictionary from the estimated channel vectors and K-SVD
based dictionaries of UEs. Before introducing the framework proposed, in Fig.
4 we provided a diagram showing the flow of the analysis described in the
paper. This diagram guides the reader through the paper.

###  III-A Common Dictionary Learning Framework

The main goal of the proposed CDL framework is to construct a CD, denoted by
$\bm{\Psi}_{c}$, that can exploit the correlation of channels across all the
UEs and BS, for improving the CSI reconstruction at the BS. The matrix of
training channel vectors is denoted by $\mathbf{H}^{\prime}$, which consists
of $M^{\prime}$ channel vectors collected for $N$ different frames across
$N_{c}$ subcarriers and $K$ UEs. Then we have $M^{\prime}=N\times N_{c}\times
K$.

To elaborate further, $\mathbf{H}^{\prime}$ is structured as
$\mathbf{{H}^{\prime}}=[\mathbf{H}^{\prime}_{1},\ldots,\mathbf{H}^{\prime}_{k},\ldots,\mathbf{H}^{\prime}_{K}]$,
and each sub-matrix in $\mathbf{H}^{\prime}$ is represented as
$\mathbf{H}^{\prime}_{k}=[\mathbf{H}^{\prime}_{1,k},\ldots,\mathbf{H}^{\prime}_{l,k},\ldots,\mathbf{H}^{\prime}_{N_{c},k}]$
$\forall k\in\\{1,2,\ldots,K\\}$, $\forall l\in\\{1,2,\ldots,N_{c}\\}$.
Similarly, $\mathbf{H}^{\prime}_{l,k}$ is defined as
$\mathbf{H}^{\prime}_{l,k}=[{\mathbf{h}}_{l_{1},k},\ldots,\mathbf{h}_{l_{n},k},\ldots,{\mathbf{h}}_{l_{N},k}]$,
where $\mathbf{h}_{l_{n},k}$ represents the channel vector transformation of
$\mathbf{H}_{l_{n},k}$ at the $n$-th MU-MIMO-OFDM frame (time-instant). We
assume that the channel envelope remains constant for an OFDM frame and then
changes for the successive frames, according to the vehicular velocity. Hence
the consecutive frames are correlated.

The sparse representation of the matrix $\mathbf{H}^{\prime}$ is denoted as
$\tilde{\mathbf{H}}^{\prime}=[\tilde{\mathbf{H}}^{\prime}_{1},\ldots,\tilde{\mathbf{H}}^{\prime}_{k},\ldots,\tilde{\mathbf{H}}^{\prime}_{K}]$,
and each sub-matrix in $\tilde{\mathbf{H}}^{\prime}$ is represented as
$\tilde{\mathbf{H}}^{\prime}_{k}=[\tilde{\mathbf{H}}^{\prime}_{1,k},\ldots,\tilde{\mathbf{H}}^{\prime}_{l,k},\ldots,\tilde{\mathbf{H}}^{\prime}_{N_{c},k}]$
$\forall k\in\\{1,2,\ldots,K\\}$, $\forall l\in\\{1,2,\ldots,N_{c}\\}$.
Similarly
$\tilde{\mathbf{H}}^{\prime}_{l,k}=[\tilde{\mathbf{h}}_{l_{1},k},\ldots,\tilde{\mathbf{h}}_{l_{n},k},\ldots,\tilde{\mathbf{h}}_{l_{N},k}]$,
where $\mathbf{\tilde{h}}_{l_{n},k}$ denotes the sparse representation of the
channel vector $\mathbf{h}_{l_{n},k}$.

The CDL optimization problem is formulated as:

$\displaystyle\min\limits_{\bm{\Psi},\tilde{\mathbf{H}}^{\prime}}{\\{\|{\mathbf{H}^{\prime}}-{\bm{\Psi}}\tilde{\mathbf{H}}^{\prime}\|_{F}^{2}\\}}\quad\\!\\!$
$\displaystyle
s.t.~{}{\|\tilde{\mathbf{h}}_{l_{n},k}\|_{{0}}\leq}~{}{S},~{}\forall~{}n\in\\{1,2,\ldots,N\\},\quad\\!\\!$
$\displaystyle~{}\forall~{}l\in\\{1,2,\ldots,N_{c}\\},~{}\forall~{}k\in\\{1,2,\ldots,K\\}.$
(8)

To solve the optimization problem in (8) we propose the following methods.

####  III-A1 CDL-KSVD method

In the CDL-KSVD method the training set $\mathbf{H}^{\prime}$ consists of the
channel vectors of all the UEs. The training set is employed to learn the
dictionary $\bm{\Psi}_{c}$ using the K-SVD algorithm[36]. The K-SVD algorithm
has two stages: the sparse coding stage and the dictionary update stage. In
the sparse coding stage, each column of ${\mathbf{H}}^{\prime}$ is sparsely
represented using a dictionary. The dictionary update stage involves updating
each column of $\bm{\Psi}$ with a dominant singular vector. As a result, the
learned dictionary $\bm{\Psi}_{c}$ has unit-norm columns.

  * •

Sparse coding stage: In the first stage of the K-SVD algorithm, the
optimization problem is formulated as:

$\displaystyle\min\limits_{\tilde{\mathbf{H}}^{\prime}}{\\{\|{\mathbf{H}^{\prime}}-{\bm{\Psi}}\tilde{\mathbf{H}}^{\prime}\|_{F}^{2}\\}}\quad\\!\\!$
$\displaystyle
s.t.~{}{\|\tilde{\mathbf{h}}_{l_{n},k}\|_{{0}}\leq}~{}{S},~{}\forall~{}n\in\\{1,2,\ldots,N\\},\quad\\!\\!$
$\displaystyle~{}\forall~{}l\in\\{1,2,\ldots,N_{c}\\},~{}\forall~{}k\in\\{1,2,\ldots,K\\}.\quad$
(9)

To solve (9), we begin by initializing the matrix $\bm{\Psi}$ with a DFT
dictionary. The next step involves finding the matrix ${\mathbf{H}}^{\prime}$
having a sparse representation, which is an $\ell_{0}$ problem and it is
carried out by using the OMP algorithm [8]. The objective of the OMP algorithm
is to find a sparse representation of ${\mathbf{H}}^{\prime}$ using a small
number of non-zero elements in the matrix $\tilde{\mathbf{H}}^{\prime}$.

  * •

Dictionary update stage: In the second stage of the K-SVD algorithm, the
optimization problem is formulated as:

$\displaystyle\min\limits_{{\bm{\Psi}}}{\\{\|{\mathbf{H}^{\prime}}-{\bm{\Psi}}\tilde{\mathbf{H}}^{\prime}\|_{F}^{2}\\}}.\quad$
(10)

The solution to the problem posed in (10) is obtained by updating each column
of the dictionary by computing a partial SVD of a matrix [6]. After the
dictionary update stage, the dictionary $\bm{\Psi}$ gets updated to
$\bm{\mathbf{\Psi}}_{c}$. It is to be noted that this method updates only the
columns corresponding to sparse coefficients of the channel matrix
$\tilde{\mathbf{H}}^{\prime}$.

  * •

Repeat sparse coding and dictionary update stages until the stopping criterion
is met.

The main advantage of the CDL-KSVD method is that it captures the spatial
correlation of the channel vectors. But its drawback is that it requires a
partial SVD operation for each column update in the dictionary update stage,
which is computationally expensive.

####  III-A2 CDL-OP method

The CDL-OP method stands for common dictionary learning - orthogonal
Procrustes method. In this method we solve the orthogonal Procrustes problem
to learn the CD [37]. The CD obtained is a square matrix with dimensions
$N_{r}N_{t}\times N_{r}N_{t}$. The constraint imposed is that the columns of
the dictionary should be orthogonal. The optimization problem is formulated
as:

$\displaystyle\min\limits_{\bm{\Psi}}{\\{\|{\mathbf{H}^{\prime}}-{\bm{\Psi}}\tilde{\mathbf{H}}^{\prime}\|_{F}^{2}\\}}\quad
s.t.~{}~{}\bm{\Psi}^{H}\bm{\Psi}=\mathbf{I},\quad$ (11)

where $\bm{\Psi}$ in (11) may be found explicitly by singular value
decomposition (SVD).

Let $\displaystyle\mathbf{C}=\tilde{\mathbf{H}}^{\prime}\mathbf{H}^{\prime H}$
$\displaystyle[\mathbf{U},\bm{\Sigma},\mathbf{V}]=\textrm{SVD}(\mathbf{C})$
$\displaystyle\bm{\Psi}=\mathbf{V}\mathbf{U}^{H}.$ (12)

The resulting dictionary $\bm{\Psi}$ obtained in (12) is the CD
($\bm{\mathbf{\Psi}}_{c}$). The main advantage of the CDL-OP method is that it
captures the spatial correlation of the channel vectors, but at the cost of an
SVD operation.

Figure 5: Overview of the CDL framework for FDD massive MU-MIMO-OFDM system.
Right: $K$ UEs with $N_{r}$ receive-antennas each; Left: massive MIMO base
station with $N_{t}$ transmit antennas; Center: For simplicity, only
dictionary feedback is shown.

###  III-B Common Dictionary for Wideband Multi-UE System

A wideband channel has a broader signal bandwidth than the coherence
bandwidth. In this section, we first discuss the CDL framework conceived for a
wideband system supporting $K$ UEs, and then highlight the simplified
scenario, where a single UE is present. Next, we will quantify the memory
savings of storing only a single CD. Then in the final sub-section, we
elaborate on the dictionary feedback reduction by sending only a single CD in
a single-UE system.

In the multi-UE system, since no communication takes place among the $K$ UEs,
CDL is impossible at any UE. Consequently, the CDL is only feasible at the BS.
For the CDL at the BS, we require the subcarrier dictionaries and the
reconstructed channels (used as training channel vectors). Let the
reconstructed sparse channel matrix at the BS be represented by
$\hat{\tilde{\mathbf{H}}}^{\prime}=[\hat{\tilde{\mathbf{H}}}^{\prime}_{1},\ldots,\hat{\tilde{\mathbf{H}}}^{\prime}_{k},\ldots,\hat{\tilde{\mathbf{H}}}^{\prime}_{K}]$,
where each sub-matrix in $\hat{\tilde{\mathbf{H}}}^{\prime}$ is represented as
$\hat{\tilde{\mathbf{H}}}^{\prime}_{k}=[\hat{\tilde{\mathbf{H}}}^{\prime}_{1,k},\ldots,\hat{\tilde{\mathbf{H}}}^{\prime}_{l,k},\ldots,\hat{\tilde{\mathbf{H}}}^{\prime}_{N_{c},k}]$
$\forall k\in\\{1,2,\ldots,K\\}$, with
$\hat{\tilde{\mathbf{H}}}^{\prime}_{l,k}=[\hat{\tilde{\mathbf{h}}}_{l_{1},k},\ldots,\hat{\tilde{\mathbf{h}}}_{l_{n},k},\ldots,\hat{\tilde{\mathbf{h}}}_{l_{N},k}]$,
containing $N$ reconstructed sparse channel vectors of each subcarrier, i.e.,
$\forall l\in\\{1,2,\ldots,N_{c}\\}$.

Let the reconstructed matrix of training channel vectors at the BS be
represented by
$\hat{\mathbf{{H}}}^{\prime}=[\hat{\mathbf{H}}^{\prime}_{1},\ldots,\hat{\mathbf{H}}^{\prime}_{k},\ldots,\hat{\mathbf{H}}^{\prime}_{K}]$,
where each sub-matrix in $\hat{\mathbf{{H}}}^{\prime}$ is represented as
$\hat{\mathbf{H}}^{\prime}_{k}=[\hat{\mathbf{H}}^{\prime}_{1,k},\ldots,\hat{\mathbf{H}}^{\prime}_{l,k},\ldots,\hat{\mathbf{H}}^{\prime}_{N_{c},k}]$
$\forall k\in\\{1,2,\ldots,K\\}$, with
$\hat{\mathbf{H}}^{\prime}_{l,k}=[\hat{\mathbf{h}}_{l_{1},k},\ldots,\hat{\mathbf{h}}_{l_{N},k}]$
when considering $N$ reconstructed channel vectors for each subcarrier i.e.,
$\forall l\in\\{1,2,\ldots,N_{c}\\}$. The total number of reconstructed
training channel vectors in $\hat{\mathbf{{H}}}^{\prime}$ is
$M^{\prime}=N\times N_{c}\times K$.

Importantly, at this stage we have to consider $\hat{\mathbf{H}}^{\prime}$
instead of $\mathbf{H}^{\prime}$ and $\hat{\tilde{\mathbf{H}}}^{\prime}$
instead of $\tilde{\mathbf{H}}^{\prime}$ in (8). The optimization problem of
finding $\bm{\Psi}_{c}$ in the multi-UE system is formulated as follows:

$\displaystyle\min\limits_{\bm{\Psi},\hat{\tilde{\mathbf{H}}}^{\prime}}{\\{\|\hat{\mathbf{H}}^{\prime}-{\bm{\Psi}}\hat{\tilde{\mathbf{H}}}^{\prime}\|_{F}^{2}\\}}\quad\\!\\!$
$\displaystyle
s.t.~{}{\|\hat{\tilde{\mathbf{h}}}_{l_{n},k}\|_{{0}}\leq}~{}{S},~{}\forall~{}n\in\\{1,2,\ldots,N\\},\quad\\!\\!$
$\displaystyle~{}\forall~{}l\in\\{1,2,\ldots,N_{c}\\},~{}\forall~{}k\in\\{1,2,\ldots,K\\}.$
(13)

The single-UE system ($K$ = 1) is a special case of a multi-UE system. For the
current system when compared to the multi-UE system of Fig. 5 there is no need
to send the subcarrier dictionaries ($\bm{\Psi}^{k}_{ksvd,l}$,
$\forall~{}n\in\\{1,2,\ldots,N\\},~{}\forall~{}l\in\\{1,2,\ldots,N_{c}\\}$)
from the UEs to the BS. The CD is learned at the UE itself using one of the
two proposed methods and then the UE sends $\bm{\Psi}_{c}$ to the BS. Now both
the BS and the UE will start using $\bm{\Psi}_{c}$. The total number of
training channel vectors in $\mathbf{H}^{\prime}$ is $M^{\prime}=N\times
N_{c}$.

The optimization problem of finding $\bm{\Psi}_{c}$ is as follows:

$\displaystyle\min\limits_{\bm{\Psi},\tilde{\mathbf{H}}^{\prime}}{\\{\|\mathbf{H}^{\prime}-{\bm{\Psi}}\tilde{\mathbf{H}}^{\prime}\|_{F}^{2}\\}}\quad\\!\\!$
$\displaystyle
s.t.~{}{\|\tilde{\mathbf{h}}_{l_{n}}\|_{{0}}\leq}~{}{S},~{}\forall~{}n\in\\{1,2,\ldots,N\\},\\!\\!$
$\displaystyle~{}\forall~{}l\in\\{1,2,\ldots,N_{c}\\}.\quad$ (14)

The CDL framework of our multi-UE system is outlined in Algorithm 1 and its
block diagram is shown in Fig. 5. More specifically, in the Algorithm 1 we
present the step-by-step procedure of the CDL framework. The remainder of this
sub-section introduces each of the steps numbered in the Fig. 5 and the
corresponding steps in the algorithm.

1

The BS sends the pilots to all the $K$-UEs in the system and each UE estimates
its channels and follows the steps 4 to 10 in the algorithm for $N$ frames and
learns the K-SVD based subcarrier dictionaries.

2

Each UE sends the $N_{c}$ K-SVD dictionaries to the BS. For the next $N$
frames each UE compresses the $l$-th subcarrier FDCHTF using the K-SVD
dictionary $(\bm{\mathbf{\Psi}}^{k}_{ksvd,l})$ and sends it to the BS. Then
the BS reconstructs the $l$-th subcarrier FDCHTF with the aid of the same
K-SVD dictionary. Using the reconstructed FDCHTFs and the K-SVD based
subcarrier dictionaries of all the $K$ UEs, the BS learns the
$\bm{\mathbf{\Psi}}_{c}$. The 2 procedure corresponds to the steps 14 to 19 in
the algorithm.

3

After learning the CD at the BS, the BS sends the CD to all the $K$-UEs, the
UEs and the BS will follow the steps 22 to 25 in the algorithm for FDCHTF
compression and reconstruction using $\bm{\mathbf{\Psi}}_{c}$.

Input: $\bm{\Psi}_{DFT}$, Pilots

1

Output: $\bm{\Psi}_{c}$

2

3Variables: Frame index $n$, UE index $k$, subcarrier index $l$.

4Implementation: $\forall k\in\\{1,\ldots,K\\}$, $\forall
l\in\\{1,\ldots,N_{c}\\}$

5for  _$n$ = $1$ to $N$_ do

6 BS transmits pilot sequences for UE1 to UEK.

7

8 The $k$-th UE estimates its channel matrix $\mathbf{H}_{l_{n},k}$ across
$l$-th subcarrier from the pilots.

9

10 After transformatation of $\mathbf{H}_{l_{n},k}$ to vector
${\mathbf{h}_{l_{n},k}}$, $k$-th UE at $l$-th subcarrier computes its
compressed vector $\mathbf{h}_{c,l_{n}}^{k}$ using ${\mathbf{h}_{l_{n},k}}$,
$\tilde{\mathbf{h}}_{l_{n},k}$, $\bm{\Psi}_{DFT}$ and sends
$\mathbf{h}_{c,l_{n}}^{k}$ to BS.

11

12 BS reconstructs $\hat{\tilde{\mathbf{h}}}_{l_{n},k}$ and
$\hat{\mathbf{h}}_{l_{n},k}$ using $\bm{\Psi}_{DFT}$ and
$\mathbf{h}_{c,l_{n}}^{k}$. BS uses $\hat{\mathbf{h}}_{l_{n},k}$ for
precoding.

13

14 In parallel $k$-th UE at $l$-th subcarrier collects training channel
vectors ${\mathbf{h}_{l_{n},k}}$’s to form matrix
$[\mathbf{h}_{l_{1},k},\ldots,\mathbf{h}_{l_{N},k}]$.

15 if  _$n$ = $N$_ then

16 The $k$-th UE learns $\bm{\Psi}_{ksvd,l}^{k}$ from
$[\mathbf{h}_{l_{1},k},\ldots,\mathbf{h}_{l_{N},k}]$ and $\bm{\Psi}_{DFT}$
using K-SVD algorithm and sends $\bm{\Psi}_{ksvd,l}^{k}$(once) to BS.

17 end if

18

19 end for

20

21for  _$n$ = $N+1$ to $2N$_ do

22 Follow steps $4$ and $5$ for another $N$ frames.

23

24 The $k$-th UE at $l$-th subcarrier computes compressed vector
$\mathbf{h}_{c,l_{n}}^{k}$ using $\mathbf{h}_{l_{n},k}$,
$\tilde{\mathbf{h}}_{l_{n},k}$, $\bm{\Psi}_{ksvd,l}^{k}$ and sends
$\mathbf{h}_{c,l_{n}}^{k}$ to BS.

25

26 BS reconstructs $\hat{\tilde{\mathbf{h}}}_{l_{n},k}$,
$\hat{\mathbf{h}}_{l_{n},k}$ using $\bm{\Psi}_{ksvd,l}^{k}$,
$\mathbf{h}_{c,l_{n}}^{k}$.

27

28 if  _$n$ = $2N$_ then

29 BS computes $\bm{\Psi}_{c}$ using one of the two methods from
$\hat{\mathbf{h}}_{l_{n},k}$’s, $\hat{\tilde{\mathbf{h}}}_{l_{n},k}$’s and
$\bm{\Psi}_{ksvd,l}^{k}$ of all UEs and sends $\bm{\Psi}_{c}$ to all the UEs.

30

31 end if

32

33 end for

34

35for  _$i >2N$_ do

36

37 Follow steps $4$ and $5$

38

39 The $k$-th UE starts using $\bm{\Psi}_{c}$ instead of
$\bm{\Psi}_{ksvd,l}^{k}$ for step $15$ and sends compressed vector
$\mathbf{h}_{c,l_{n}}^{k}$ to BS.

40

41 BS reconstructs $\hat{\tilde{\mathbf{h}}}_{l_{n},k}$ and
$\hat{\mathbf{h}}_{l_{n},k}$ of $k$-th UE using $\bm{\Psi}_{c}$ and
$\mathbf{h}_{c,l_{n}}^{k}$(obtained from step $23$).

42

43 BS uses $\hat{\mathbf{h}}_{l_{n},k}$ of each UE for precoding.

44

45 end for

46

Algorithm 1 CDL Framework for $K$ Users

####  III-B1 Memory Reduction Calculation

  * •

The memory required to store each subcarrier dictionary
$\bm{\Psi}^{k}_{ksvd,l}$ is $(N_{r}N_{t})^{2}$, $\forall
l\in\\{1,2,\ldots,N_{c}\\},\forall k\in\\{1,2,\ldots,K\\}$.

  * •

The total memory required to store $N_{c}$ subcarrier dictionaries is
$N_{c}(N_{r}N_{t})^{2}$.

  * •

The memory required to store $\bm{\Psi}_{c}$ is $(N_{r}N_{t})^{2}$.

  * •

Total memory storage reduction for a $K$-UE system $=$ $K\times$ ([Memory
required to store $N_{c}$ subcarrier dictionaries] $-$ [Memory required to
store $\bm{\Psi}_{c}]$), which is formulated as:

$\displaystyle\Delta_{saved}$ $\displaystyle=K(N_{c}-1)(N_{r}N_{t})^{2}.$ (15)

  * •

The total memory storage reduction for a single-UE is

$\displaystyle\Delta_{saved}$ $\displaystyle=(N_{c}-1)(N_{r}N_{t})^{2}.$ (16)

####  III-B2 Dictionary Feedback Reduction Calculation by Sending a CD in a
Single-UE System

  * •

The dimension of each subcarrier K-SVD dictionary $\bm{\Psi}_{ksvd,l}$ is
$N_{r}N_{t}\times N_{r}N_{t}$, $\forall l\in\\{1,2,\ldots,N_{c}\\}$.

  * •

The total dimension of $N_{c}$ subcarrier dictionaries is $N_{c}\times
N_{r}N_{t}\times N_{r}N_{t}$.

  * •

The dimension of $\bm{\Psi}_{c}$ is $N_{r}N_{t}\times N_{r}N_{t}$.

  * •

Total dictionary feedback reduction for a single-UE ($\mathcal{T}_{saved}$)
$=$ [Feedback required for sending $N_{c}$ subcarrier dictionaries
($\mathcal{T}_{ksvd}$)] $-$ [Feedback required for sending $\bm{\Psi}_{c}$
($\mathcal{T}_{com}$)], where we have

$\displaystyle\mathcal{T}_{ksvd}=N_{c}(N_{r}N_{t})^{2}\quad\\!\\!$
$\displaystyle\mathcal{T}_{com}=(N_{r}N_{t})^{2}\quad\\!\\!$
$\displaystyle\mathcal{T}_{saved}=\mathcal{T}_{ksvd}-\mathcal{T}_{com}=(N_{c}-1)(N_{r}N_{t})^{2}.\quad\\!\\!$

  * •

Reduction in dictionary feedback: We define the feedback reduction factor by
($\Upsilon$):

$\Upsilon=\displaystyle\frac{\mathcal{T}_{com}}{\mathcal{T}_{ksvd}}\\\
=\displaystyle\frac{1}{N_{c}}$ (17)

TABLE V: Dictionary feedback reduction factor in a single-UE System. $N_{t}$ | $N_{r}$ | $N_{c}$ | $\mathcal{T}_{ksvd}$ | $\mathcal{T}_{com}$ | $\Upsilon$
---|---|---|---|---|---
$64$ | $1$ | $4$ | $16384$ | $4096$ | $1/4$
$64$ | $1$ | $32$ | $131072$ | $4096$ | $1/32$

####  III-B3 Computational Complexity of the Algorithm

We calculate the computational complexity of the dictionary learning stage for
both the CDL-OP and the CDL-KSVD methods.

a) CDL-OP method:

  * •

In (12), the SVD operation requires all the eigenvectors, resulting in a full
SVD operation.

  * •

The computation of a full SVD operation, specifically using the Golub-Reinsch
SVD (GR-SVD) method, requires 21$(N_{r}N_{t})^{3}$ floating-point operations
(FLOPS) [6]. On the other hand, the Chan-SVD (R-SVD) method requires
26$(N_{r}N_{t})^{3}$ FLOPS [6].

  * •

Therefore, to update the dictionary, we require a computational complexity of
$\mathcal{O}[(N_{r}N_{t})^{3}]$.

b) CDL-KSVD method:

  * •

In (10), updating each column of the dictionary requires an SVD operation.
This SVD operation only requires the dominant eigenvector, resulting in a
partial SVD computation.

  * •

The GR-SVD method requires $14N_{r}N_{t}N_{c}^{\prime 2}+9N_{c}^{\prime 3}$
FLOPS [6], while the R-SVD method requires $6N_{r}N_{t}N_{c}^{\prime
2}+20N_{c}^{\prime 3}$ FLOPS [6]. Here, $N_{c}^{\prime}$ represents the number
of non-zero coefficients corresponding to the $k$-th row in
$\tilde{\mathbf{H}}^{\prime}$, and $N_{c}^{\prime}$ ranges from 0 to
$M^{\prime}$.

  * •

Therefore, to update the complete dictionary, a total of $N_{r}N_{t}$ partial
SVD operations are required.

c) For example, let us consider $N_{t}=64$, $N_{r}=1$, $M^{\prime}=1600$, and
an average value of $N_{c}^{\prime}=M^{\prime}/4$. The computational
complexity in FLOPS is provided in Table VI. We represent the number of FLOPS
required for updating a single column in the dictionary by CDL-KSVD (min), and
that imposed by updating all columns in the dictionary using CDL-KSVD (max).

TABLE VI:  Computational complexity in FLOPS Method | GR-SVD | Chan-SVD
---|---|---
CDL-OP | $5.505*10^{6}$ | $6.8157*10^{6}$
CDL-KSVD (min) | $7.1936*10^{8}$ | $1.3414*10^{9}$
CDL-KSVD (max) | $4.6039*10^{10}$ | $8.5852*10^{10}$

####  III-B4 CSI Feedback Case Study in a Single UE

The dimension of the compressed channel vector $\mathbf{h}^{k}_{c,l}$
($\in\mathbb{C}^{N_{g}\times 1}$) sent from the UE in the uplink can be varied
by adjusting the compression factor $g$. Specifically, the dimension is given
by $N_{g}=\frac{N_{r}N_{t}}{g}$. By tuning the value of $g$, we can
beneficially reduce the amount of CSI feedback required.

To quantify the feedback requirements, we introduce the variables $\gamma_{u}$
and $\gamma_{c}$ to represent the feedback for the non-dictionary and
dictionary-based methods, respectively. In a non-dictionary based method
without compression, the CSI fed back from the UE corresponds to
$N_{c}N_{r}N_{t}$ elements for one frame. For $N^{\prime}$ frames the CSI
feedback will be $\gamma_{u}$ = $N^{\prime}N_{c}N_{r}N_{t}$. However, in a
dictionary-based method associated with compression, the feedback is
constituted by the CSI information having $\frac{N_{c}N_{r}N_{t}}{g}$ elements
for one frame, along with a one-time transmission of a dictionary with
$(N_{r}N_{t})^{2}$ elements. So, for a total of $N^{\prime}$ frames the
feedback is given by $\gamma_{c}$ = $(N_{r}N_{t})^{2}$ \+
$\frac{N^{\prime}N_{c}N_{r}N_{t}}{g}$.

We define $\Gamma$ as the CSI feedback ratio, which is calculated as the ratio
between ${\gamma_{c}}$ and ${\gamma_{u}}$. If $\Gamma<1$, it indicates that
the value of $\gamma_{c}$ is lower than $\gamma_{u}$, resulting in a saving in
CSI feedback.

${\Gamma=\frac{\gamma_{c}}{\gamma_{u}}=\frac{(N_{r}N_{t})^{2}+\frac{N^{\prime}N_{c}N_{r}N_{t}}{g}}{N^{\prime}N_{c}N_{r}N_{t}}=\frac{N_{r}N_{t}}{N^{\prime}N_{c}}+\frac{1}{g}}$
(18)

For example in Table VII we consider a scenario associated with
$N^{\prime}=2^{10}$ and vary the values of $N_{c}$, $N_{r}$, and $g$. Using
the formula given in 18, we demonstrate significant reductions in CSI
feedback. The level of compression applied to the channel vector
$\mathbf{h}^{k}_{c,l}$ depends on the sparsity parameter $S$. As per the lower
bound, the number of elements in $\mathbf{h}^{k}_{c,l}$ must satisfy
$N_{g}>2S$ [9]. In Fig. 13, we illustrate the impact of the compression factor
$g$ on the BER vs. signal-to-noise ratio (SNR) performance for both the non-
dictionary and dictionary-based methods.

TABLE VII: CSI feedback savings comparison table for $N^{\prime}$ = $2^{10}$. $g$ | $N_{c}$ | $N_{t}$ | $N_{r}$ |  $\Gamma$ = $\frac{\gamma_{c}}{\gamma_{u}}$
---|---|---|---|---
2 | 32 | 64 | 1 | $0.5$
2 | 64 | 64 | 2 | $0.5$
4 | 32 | 64 | 1 | 0.252
4 | 64 | 64 | 2 | 0.252

##  IV Numerical Results

In this section, we provide the simulation results for characterising the NMSE
performances of the DFT dictionary ($\bm{\Psi}_{DFT}$), of the individual
K-SVD based subcarrier dictionaries ($\bm{\Psi}^{k}_{ksvd,l}$) and of the
proposed CD ($\bm{\Psi}_{c}$). Using the DFT dictionary as the initial
reference dictionary, we obtain the individual K-SVD based subcarrier
dictionaries. Then $\bm{\Psi}_{c}$ is learned by the proposed methods. For
NMSE calculations, each dictionary is used for reconstructing the $P$ channel
vectors of each subcarrier at the BS.

The NMSE of the reconstructed channel is used as a performance metric defined
as

$\mathbf{NMSE}=\frac{1}{P}\smashoperator[r]{\sum_{i=1}^{P}}\frac{\|\hat{\mathbf{h}}_{l_{n},k}-\mathbf{h}_{l_{n},k}\|_{2}^{2}}{\|\mathbf{h}_{l_{n},k}\|_{2}^{2}},$
(19)

where $\hat{\mathbf{h}}_{l_{n},k}$ is the reconstructed channel vector and
$\mathbf{h}_{l_{n},k}$ is the original one.

###  IV-A Simulation Settings

The simulations are carried out for a massive MIMO-OFDM system having
$N_{t}=64$, antenna spacing of $d=\lambda/15$, carrier wavelength $\lambda$,
operating at a carrier frequency of $f_{c}$ = $2$ GHz. Furthermore, we have a
communication bandwidth of $B$ = $20$ MHz, $K=1$ and $K=3$ UEs, $N=50$,
$P=500$ test channel vectors, $N_{c}=32$ and $M=N_{r}N_{t}/2$. For $M=32$, all
the existing and the proposed dictionaries reduce the CSI feedback by 50% in
the uplink. For all the experiments, the subcarrier K-SVD based dictionaries
are learned for $N=50$ from each subcarrier.

For experiments in the multi-UE system, the channels are generated using a
Quasi Deterministic Radio Channel Generator (QuaDRiGa) [40, 41] for the three
UEs having velocities of $V=10$, $15$ and $20$ kmph. For experiments in the
single-UE system, the channels are taken from the UE of $V=20$ kmph. The
channel update rate (CUR) considered to generate channels in QuaDRiGa is $10$
ms. The QuaDRiGa simulation platform is recommended by 3GPP (3rd Generation
Partnership Project) for designing and simulating wireless communications
systems.

The main motivation for the CDL is not only to reduce the CSI feedback but
also to reduce the CSI reconstruction NMSE. The benefit of the proposed
methods in terms of the NMSE performance has to be studied. Hence we have
conducted experiments for determining which of the proposed methods will best
replace the DFT dictionary and subcarrier K-SVD based dictionaries in both
single-UE and multi-UE systems. Since we have considered $N_{c}=32$, it is not
feasible to show the CD performance across all the subcarriers. In the single-
UE system, we have considered subcarriers $l=1$ or $8$, since the channel
gains of these two subcarriers are relatively low over the period of time. In
the multi-UE system, we have considered the first subcarrier $l=1$ of all
three UEs to evaluate the NMSE performance as a function of sparsity and
compared the proposed methods’ CD performance to the DFT and K-SVD
dictionaries in the literature.

In OFDM systems, the need for subcarrier K-SVD based dictionaries increases
with the number of subcarriers. The FDCHTFs of each subcarrier are considered
to be independent in a wideband OFDM system, but this is only realistic for
extremely long CIRs. Hence in the proposed system we assume having realistic
correlation among subcarriers in the FD. This correlation among the
subcarriers is captured by the CD using one of the two proposed methods. Our
CDL procedure may also be extended to larger $N_{c}$.

In all the simulation results, for each subcarrier it can be observed that the
NMSE decreases as the sparsity increases. This is because in the sparse vector
transformation (5) the sparse vector $\tilde{\mathbf{h}}$ picks many columns
in the dictionary for higher sparsities, which in turn helps to solve the
optimization problem (7) by minimizing the distance between $\mathbf{h}$ and
$\tilde{\mathbf{h}}$. Hence, higher sparsity will improve the reconstruction
performance.

######  Experiment 1.

In this single-UE experiment, we study the NMSE performance across a
particular subcarrier of the UE using the CD learned from the proposed methods
and existing methods as a function of sparsity. The proposed methods’ CD is
learned across all the subcarriers of the UE. For this experiment we consider
a massive SU-MIMO-OFDM system having $N_{c}=32$ subcarriers and a UE having
$N_{r}=1$ RA and moving with a velocity of 20 kmph.

Figure 6: A single-UE system is considered, where the NMSE performance
comparison among all the dictionaries is carried out for $N_{t}$ = 64, $N_{r}$
= 1 and $N_{c}$ = 32. Subcarrier 1 is considered for comparison and the UE
velocity is 20 kmph.

In Fig. 6, we consider the subcarrier $l=1$ to study the NMSE performance of
all the dictionaries. For a particular sparsity index of $S=8$, the NMSE of
the CDL-KSVD dictionary is $1.7\times 10^{-2}$, of the CDL-OP dictionary is
$1.3\times 10^{-2}$, of the subcarrier K-SVD based dictionary is $1.8\times
10^{-2}$, and of the DFT dictionary is $3.3\times 10^{-2}$. The CDL-OP
dictionary has the lowest NMSE at $S=8$. For all other sparsities, the CDL-OP
and the CDL-KSVD method dictionaries perform similarly, and the NMSE values
are close to those of the subcarrier K-SVD dictionary. All the proposed
methods’ CD exhibit better performance than the DFT dictionary.

Figure 7: A single-UE system is considered, where the NMSE performance
comparison among the CDL-OP dictionaries is carried out for $N_{t}$ = 16, 32
and 64, $N_{r}$ = 1 and $N_{c}$ = 32. Subcarrier 1 is considered for
comparison and the UE velocity is 20 kmph.

In Fig. 7, we consider a scenario where the BS has a different number of
antennas, namely $N_{t}$ = 16, 32, and 64. We consider the subcarrier $l=1$ to
study the NMSE performance of the CDL-OP dictionaries learned for different
$N_{t}$ values. We observe that the NMSE value increases with the number of
antennas at the BS.

######  Experiment 2.

In this single-UE experiment, we study the NMSE performance across different
subcarriers of the UE employing the CD learned by the proposed methods and the
individual subcarrier K-SVD based dictionaries as a function of sparsity. For
this experiment we consider a massive SU-MIMO-OFDM system having $N_{c}=32$
subcarriers. The UE is equipped with $N_{r}=1$ (and $2$) RAs and is moving at
a velocity of 20 kmph.

Figure 8: A single-UE system is considered, where the NMSE performance of the
K-SVD dictionary and the CDL-KSVD dictionary are used for $N_{t}$ = 64,
$N_{r}$ = 1 and $N_{c}$ = 32. Subcarriers 1 and 8 are considered for
comparison and the UE velocity is 20 kmph. Figure 9: A single-UE system is
considered, where the NMSE performance of the K-SVD dictionary and the CDL-OP
dictionary are used for for $N_{t}$ = 64, $N_{r}$ = 1 and $N_{c}$ = 32.
Subcarriers 1 and 8 are considered for comparison and the UE velocity is 20
kmph.

In Fig. 8, the $\bm{\Psi}_{c}$ employed for NMSE characterization is learned
from the CDL-KSVD method and in Fig. 9, the $\bm{\Psi}_{c}$ employed for NMSE
characterization is learned from the CDL-OP method. Observe from Figs. 8 and
9, for subcarriers $l=1$ and $8$ at all the sparsity index values, the NMSE
values are close to each other, but the CDL-OP method is the best among all
the three methods in terms of the NMSE performance attained. Both the CDL-KSVD
and CDL-OP methods rely on the SVD operation and learn the CD from the
channels estimated at the UE. Consequently, the NMSE reconstruction results
shown in Figs. 8 and 9 exhibit a high degree of similarity.

We have carried out a simulation and presented the results in Fig. 10, where
we specifically focus on the scenario where $N_{r}>1$, indicating the presence
of multiple receive antennas. By considering this scenario, we ensure that our
analysis is not limited to a specific number of receive antennas. It is
observed that all the proposed methods’ CD exhibit better NMSE performance
than the DFT dictionary.

Figure 10: A single-UE system with multiple UE antennas is considered, where
the NMSE performance comparison among all the dictionaries is carried out for
$N_{t}$ = 64, $N_{r}$ = 2 and $N_{c}$ = 32. Subcarrier 1 is considered for
comparison and the UE velocity is 20 kmph.

######  Experiment 3.

In this multi-UE experiment, we initially study the NMSE performance of the
dictionary generated using the CDL-OP method and existing methods for a
particular UE’s subcarrier as a function of sparsity. Then we study the NMSE
performance of the CDL-OP dictionary across different UEs for a particular
subcarrier. The CDL-OP dictionary is learned across all the subcarriers and
all the $K$ UEs. For the experiment, we consider a massive MU-MIMO-OFDM system
having $N_{c}=32$ subcarriers and $K=3$ UEs each associated with $N_{r}=1$ RA.

Figure 11: A multi-UE system is considered, where the NMSE performance
comparison of the DFT, the K-SVD and the CDL-OP methods is carried out for $K$
= 3, $N_{t}$ = 64, $N_{r}$ = 1 and $N_{c}$ = 32. Subcarrier 1 of UE1 is
considered for comparison. The velocities of the three UEs are 10, 15, and 20
kmph.

The wireless channels are generated for three UEs having velocities of $V=$
10, 15, and 20 kmph using a QuaDRiGa simulator. Let us assume that a UE
changes its velocity to that of another UE, but experiences different channel
characteristics. In that case, there is no need to learn a new dictionary for
that particular UE. The CD has already captured all the three UE channel
characteristics, and this procedure can be extended to larger $K$ and $N_{c}$
values.

In Fig. 11, we consider the first UE and subcarrier $l=1$ to study the NMSE
performance of the dictionaries from the full set of $N_{c}=32$ subcarriers
and $K=3$ UEs. For a particular sparsity of $S=8$, the NMSE value of the CDL-
OP dictionary is $1.3\times 10^{-2}$, of the subcarrier K-SVD based dictionary
is $1.8\times 10^{-2}$, and of the DFT dictionary is $3.3\times 10^{-2}$. We
observe that all the proposed methods’ CD have better NMSE performance than
the DFT dictionary. Among the proposed methods, the CDL-KSVD method exhibits
the poorest NMSE performance. Therefore, we are not pursuing this method any
further in the simulations.

Figure 12: In a multi-UE system, the NMSE performance of the K-SVD dictionary
and CDL-OP method’s dictionary for $K$ = 3, $N_{t}$ = 64, $N_{r}$ = 1 and
$N_{c}$ = 32. Subcarrier 1 of UE2 and UE3 are considered for comparison.
Velocities of three UEs are 10, 15, and 20 kmph.

In Fig. 12, the $\bm{\Psi}_{c}$ employed for NMSE characterization is learned
from the CDL-OP method. Observe from the Fig. 12, for subcarrier $l=1$ of UE2
and UE3 at almost all the sparsity index values, the NMSE values for the
proposed CDL-OP method’s dictionary is better than subcarrier K-SVD based
dictionaries, because the CDL-OP dictionary is learned from the estimated
channels of K-SVD based dictionaries.

######  Experiment 4.

In this single-UE experiment, we study the BER performance at the UE as a
function of the SNR. The data symbols are transmitted from the BS to the UE in
two ways: a) Using the true channel estimates without compression and b) Using
the channel estimates with compression that are obtained from the CDL-OP
dictionary.

Figure 13: In a single-UE system with 64 TAs at the BS and a UE with 1 RA, we
examine the BER performance of channel estimates using the CDL-OP methods’
dictionary. We consider two scenarios: $N_{g}=N_{r}N_{t}/2$ and
$N_{g}=N_{r}N_{t}/4$. For comparison, we assume uncompressed channel estimates
for subcarrier 1. The UE has a velocity of 20 kmph.

To elaborate further on the performance of our proposed framework, we analyze
the BER in a downlink scenario. In Fig. 13 we evaluate the BER using two sets
of channels: the true uncompressed channels and the channels estimated using
the CDL-OP dictionary on subcarrier $1$. The precoder matrices employed at the
BS are denoted as
$\mathbf{W}^{g}=diag(\frac{{\mathbf{h}}_{1}^{H}}{|{\mathbf{h}}_{1}|})$ and
$\mathbf{W}^{g}_{op}=diag(\frac{\hat{\mathbf{h}}_{1,op}^{H}}{|\hat{\mathbf{h}}_{1,op}|})$,
which correspond to the weights obtained from the true channels and to the
channels estimated using CDL-OP dictionary, respectively. Let $\mathbf{x}$
represents the modulated symbol vector. Explicitly, in our BER analysis, we
harness a half-rate convolutional encoder having the generator sequences of G
= [101, 111] and the resultant bits are 16-PSK modulated for generating the
symbol vector $\mathbf{x}$. The received signal in the case of true
uncompressed channels at the UE can be represented as
${y}=\mathbf{h}_{1}^{T}\mathbf{W}^{g}\mathbf{x}+n$, where $n$ is the additive
noise. Similarly the receieved signal for channels estimated using the CDL-OP
dictionary can be represented as
${y_{op}}=\mathbf{h}_{1}^{T}\mathbf{W}^{g}_{op}\mathbf{x}+n$.

The CDL-OP dictionaries used in this experiment are learned at a sparsity of
16 for $g=2$ and at a sparsity of 8 for $g=4$. As we increase the compression
factor $g$ to reduce the CSI feedback from the UE, we observe a BER
performance erosion. The lower bound shown in Fig. 13 represents the BER
performance when the BS employs uncompressed CSI. Notably, as depicted in Fig.
13, the BER of the UE, recorded for $g=2$ when utilizing the CDL-OP dictionary
approaches the lower bound.

##  V Summary and Conclusions

We proposed a novel CDL framework for reducing the FDCHTF feedback and memory
requirement of the UE and BS. The framework is more beneficial for the UE,
which is usually resource-constrained, and the savings can be significant. For
the simulations, the channels are generated using the QuaDRiGa simulator. In a
multi-UE system of three UEs and in a single-UE system, all the proposed
method’s dictionaries have better NMSE performance than the DFT dictionary.
The CDL-OP dictionary performs better than the CDL-KSVD dictionary for all the
sparsities. Hence the CDL-OP method can be beneficially employed for CDL to
compress the CSI and improve the CSI reconstruction performance. In terms of
the computational complexity, the CDL-OP method requires only a single SVD
operation to learn the CD, while the CDL-KSVD method requires an SVD operation
for learning each column in the CD. So the CDL-OP method has lower
computational complexity than the CDL-KSVD method. To minimize the impact
imposed on the BER performance, it is important to choose an appropriate
compression ratio $g$. In our case, a compression ratio of $g=2$ is considered
to have a modest impact on the BER performance. By selecting a suitable
compression ratio, the system can strike a balance between reducing the amount
of feedback, while maintaining a satisfactory BER performance.

We conclude by highlighting the differences between the multi-UE and single-UE
systems as follows:

  1. 1.

In the multi-UE system, the memory is reduced by a factor of $N_{c}$ at the UE
and $KN_{c}$ at the BS by having only a single CD instead of multiple
subcarrier dictionaries. For $N_{t}=64$ and $N_{c}=32$, the memory required
for storing the CD at each UE is reduced by a factor of 32 and the CSI
feedback is reduced by a factor of two.

  2. 2.

In the single-UE system, the memory is reduced by a factor of $N_{c}$ at both
the UE and BS, the dictionary feedback is also reduced by a factor of $N_{c}$
in the uplink. This is achieved by sending only a single CD instead of $N_{c}$
subcarrier dictionaries to the BS. For $N_{t}=64$ and $N_{c}=32$, the memory
is reduced by a factor of 32 for storing only a CD, and the dictionary
feedback is also reduced by a factor of 32. Finally, the CSI feedback is
reduced by a factor of two.

  3. 3.

In the multi-UE system, the CD ($\bm{\Psi}_{c}$) generated using the CDL-OP
has a better NMSE performance to that of the individual subcarrier K-SVD
dictionaries $\bm{\Psi}^{k}_{ksvd,l}$ $\forall l\in[1,N_{c}],\forall
k\in[1,K]$.

  4. 4.

In the single-UE system, the CD ($\bm{\Psi}_{c}$) generated using the CDL-OP
method has a better NMSE performance to that of the individual subcarrier
K-SVD dictionaries $\bm{\Psi}_{ksvd,l}$ $\forall l\in[1,N_{c}]$.

## References

  * [1] E. Björnson, J. Hoydis, and L. Sanguinetti, “Massive MIMO has unlimited capacity,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 1, pp. 574–590, 2018.
  * [2] S. Yang and L. Hanzo, “Fifty years of MIMO detection: The road to large-scale MIMOs,” _IEEE Commun. Surv. Tutorials._ , vol. 17, no. 4, pp. 1941–1988, 2015.
  * [3] R. Shafin and L. Liu, “Multi-cell multi-user massive FD-MIMO: Downlink precoding and throughput analysis,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 1, pp. 487–502, 2019.
  * [4] Z. Zhong, L. Fan, and S. Ge, “FDD massive MIMO uplink and downlink channel reciprocity properties: Full or partial reciprocity?” in _Proc. IEEE Global Commun. Conf. (GLOBECOM)_ , 2020, pp. 1–5.
  * [5] Z. Zhang, X. Wang, K. Long, A. V. Vasilakos, and L. Hanzo, “Large-scale MIMO-based wireless backhaul in 5G networks,” _IEEE Wireless Commun._ , vol. 22, no. 5, pp. 58–66, 2015.
  * [6] P. K. Gadamsetty and K. V. S. Hari, “A fast dictionary learning algorithm for CSI feedback in massive MIMO FDD systems,” in _Proc. National Conf. on Commun. (NCC)_ , 2023, pp. 1–6.
  * [7] S. S. Yılmaz and B. Özbek, “Compressive sensing based low complexity user selection for massive MIMO systems,” in _Proc. VTC Spring IEEE 91st Veh. Technol. Conf._ , 2020, pp. 1–6.
  * [8] J. Wen, Z. Zhou, J. Wang, X. Tang, and Q. Mo, “A sharp condition for exact support recovery with orthogonal matching pursuit,” _IEEE Trans. Signal Process._ , vol. 65, no. 6, pp. 1370–1382, 2017.
  * [9] X.-J. Liu, S.-T. Xia, and F.-W. Fu, “Reconstruction guarantee analysis of basis pursuit for binary measurement matrices in compressed sensing,” _IEEE Trans. Inf. Theory_ , vol. 63, no. 5, pp. 2922–2932, 2017.
  * [10] H. Ge, L. Wang, J. Wen, and J. Xian, “An RIP condition for exact support recovery with covariance-assisted matching pursuit,” _IEEE Signal Process. Lett._ , vol. 26, no. 3, pp. 520–524, 2019.
  * [11] L. Lu, G. Y. Li, D. Qiao, and W. Han, “Sparsity-enhancing basis for compressive sensing based channel feedback in massive MIMO systems,” in _Proc. IEEE Global Commun. Conf. (GLOBECOM)_ , 2015, pp. 1–6.
  * [12] L. Hanzo, B. Choi, T. Keller _et al._ , _OFDM and MC-CDMA for broadband multi-user communications, WLANs and broadcasting_. John Wiley & Sons, 2005.
  * [13] P. Cheng and Z. Chen, “Multidimensional compressive sensing based analog CSI feedback for massive MIMO-OFDM systems,” in _Proc. VTC Fall - IEEE 80th Veh. Technol. Conf._ , 2014, pp. 1–6.
  * [14] W. Zeng, Y. He, B. Li, and S. Wang, “Sparsity learning-based CSI feedback for FDD massive MIMO systems,” _IEEE Wireless Commun. Lett._ , vol. 10, no. 3, pp. 585–588, 2021.
  * [15] Z. Lv and Y. Li, “A channel state information feedback algorithm for massive MIMO systems,” _IEEE Commun. Lett._ , vol. 20, no. 7, pp. 1461–1464, 2016.
  * [16] S. Qiu, D. Gesbert, D. Chen, and T. Jiang, “A covariance-based hybrid channel feedback in FDD massive MIMO systems,” _IEEE Trans. Commun._ , vol. 67, no. 12, pp. 8365–8377, 2019.
  * [17] H. Xie and N. González-Prelcic, “Dictionary learning for channel estimation in hybrid frequency-selective mmWave MIMO systems,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 11, pp. 7407–7422, 2020.
  * [18] Y. Zhao, Y. Teng, A. Liu, and V. Lau, “Two-timescale joint UL/DL dictionary learning and channel estimation in massive MIMO systems,” in _Proc. IEEE Global Commun. Conf. (GLOBECOM)_ , 2022, pp. 5408–5413.
  * [19] F. Kulsoom, A. Vizziello, H. N. Chaudhry, and P. Savazzi, “Joint sparse channel recovery with quantized feedback for multi-user massive MIMO systems,” _IEEE Access_ , vol. 8, pp. 11 046–11 060, 2020.
  * [20] B. Lee, J. Choi, J.-Y. Seol, D. J. Love, and B. Shim, “Antenna grouping based feedback compression for FDD-based massive MIMO systems,” _IEEE Trans. Commun._ , vol. 63, no. 9, pp. 3261–3274, 2015.
  * [21] W. Shen, L. Dai, B. Shim, Z. Wang, and R. W. Heath, “Channel feedback based on AoD-adaptive subspace codebook in FDD massive MIMO systems,” _IEEE Trans. Commun._ , vol. 66, no. 11, pp. 5235–5248, 2018.
  * [22] G. Huang, A. Liu, and M.-J. Zhao, “Two-stage adaptive and compressed CSI feedback for FDD massive MIMO,” _IEEE Trans. Veh. Technol._ , vol. 70, no. 9, pp. 9602–9606, 2021.
  * [23] Z. Wei, H. Li, H. Liu, B. Li, and C. Zhao, “Randomized low-rank approximation based massive MIMO CSI compression,” _IEEE Commun. Lett._ , vol. 25, no. 6, pp. 2004–2008, 2021.
  * [24] Y. Liao, H. Yao, Y. Hua, and C. Li, “CSI feedback based on deep learning for massive MIMO systems,” _IEEE Access_ , vol. 7, pp. 86 810–86 820, 2019\.
  * [25] T. Wang, C.-K. Wen, S. Jin, and G. Y. Li, “Deep learning-based CSI feedback approach for time-varying massive MIMO channels,” _IEEE Wireless Commun. Lett._ , vol. 8, no. 2, pp. 416–419, 2019.
  * [26] H. Ye, F. Gao, J. Qian, H. Wang, and G. Y. Li, “Deep learning-based denoise network for CSI feedback in FDD massive MIMO systems,” _IEEE Commun. Lett._ , vol. 24, no. 8, pp. 1742–1746, 2020.
  * [27] P. Liang, J. Fan, W. Shen, Z. Qin, and G. Y. Li, “Deep learning and compressive sensing-based CSI feedback in FDD massive MIMO systems,” _IEEE Trans. Veh. Technol._ , vol. 69, no. 8, pp. 9217–9222, 2020.
  * [28] S. Tang, J. Xia, L. Fan, X. Lei, W. Xu, and A. Nallanathan, “Dilated convolution based CSI feedback compression for massive MIMO systems,” _IEEE Trans. Veh. Technol._ , vol. 71, no. 10, pp. 11 216–11 221, 2022\.
  * [29] Z. Hu, G. Liu, Q. Xie, J. Xue, D. Meng, and D. Gündüz, “A learnable optimization and regularization approach to massive MIMO CSI feedback,” _IEEE Trans. Wireless Commun._ , pp. 1–1, 2023.
  * [30] Z. Cao, J. Guo, C.-K. Wen, and S. Jin, “Deep-unfolding-based bit-level CSI feedback in massive MIMO systems,” _IEEE Wireless Commun. Lett._ , vol. 12, no. 2, pp. 371–375, 2023.
  * [31] P.-H. Kuo, H. T. Kung, and P.-A. Ting, “Compressive sensing based channel feedback protocols for spatially-correlated massive antenna arrays,” in _Proc. IEEE Wireless Commun. Netw. Conf. (WCNC)_ , 2012, pp. 492–497.
  * [32] Y. Barbotin, A. Hormati, S. Rangan, and M. Vetterli, “Estimation of sparse MIMO channels with common support,” _IEEE Trans. Commun._ , vol. 60, no. 12, pp. 3705–3716, 2012.
  * [33] L. Yang, J. Fang, and H. Li, “Sparse Bayesian dictionary learning with a Gaussian hierarchical model,” in _Proc. IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP)_ , 2016, pp. 2564–2568.
  * [34] K. Zhang, E. L. Xu, H. Zhang, Z. Feng, and S. Cui, “Data driven automatic modulation classification via dictionary learning,” _IEEE Wireless Commun. Lett._ , vol. 7, no. 4, pp. 586–589, 2018.
  * [35] M. Nazzal, M. A. Aygül, A. Görçin, and H. Arslan, “Dictionary learning-based beamspace channel estimation in millimeter-wave massive MIMO systems with a lens antenna array,” in _Proc. 15th Int. Wireless Commun. & Mobile Computing Conf. (IWCMC)_, 2019, pp. 20–25.
  * [36] M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation,” _IEEE Trans. Signal Process._ , vol. 54, no. 11, pp. 4311–4322, 2006.
  * [37] J. C. Gower, G. B. Dijksterhuis _et al._ , _Procrustes problems_. Oxford University Press on Demand, 2004, vol. 30.
  * [38] C.-K. Wen, W.-T. Shih, and S. Jin, “Deep learning for massive MIMO CSI feedback,” _IEEE Wireless Commun. Lett._ , vol. 7, no. 5, pp. 748–751, 2018\.
  * [39] M. Ettorre, S. C. Pavone, M. Casaletti, and M. Albani, “Experimental validation of Bessel beam generation using an inward hankel aperture distribution,” _IEEE Trans. Antennas and Propagation_ , vol. 63, no. 6, pp. 2539–2544, 2015.
  * [40] Z. F. et al., “Channel simulation and validation by Quadriga for suburban microcells under 6 GHz,” in _Proc. 12th Int. Symp. on Antennas, Propagation and EM Theory (ISAPE)_ , 2018, pp. 1–4.
  * [41] F. Burkhardt, S. Jaeckel, E. Eberlein, and R. Prieto-Cerdeira, “Quadriga: A MIMO channel model for land mobile satellite,” in _Proc. The 8th European Conf. on Antennas and Propagation (EuCAP 2014)_ , 2014, pp. 1274–1278.