Corresponding author contact information:
C. Garcia: E-mail<EMAIL_ADDRESS>
# A deep learning model for brain vessel segmentation
in 3DRA with arteriovenous malformations

Camila García  Yatiris Group, PLADEMA Institute, UNICEN, Campus Universitario,
Tandil, Argentina  Consejo Nacional de Investigaciones Científicas y Técnicas,
CONICET, Tandil, Argentina  Yibin Fang  Department of Neurovascular Disease,
Shanghai Fourth People’s Hospital, School of Medicine, Tongji University,
Shanghai, China  Jianmin Liu  Department of Neurosurgery, Changhai Hospital,
Naval Medical University, Shanghai, China  Ana Paula Narata  Department of
Neuroradiology, University Hospital of Southampton, Southampton, UK  José
Ignacio Orlando  Yatiris Group, PLADEMA Institute, UNICEN, Campus
Universitario, Tandil, Argentina  Consejo Nacional de Investigaciones
Científicas y Técnicas, CONICET, Tandil, Argentina  Ignacio Larrabide  Yatiris
Group, PLADEMA Institute, UNICEN, Campus Universitario, Tandil, Argentina
Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET, Tandil,
Argentina

###### Abstract

Segmentation of brain arterio-venous malformations (bAVMs) in 3D rotational
angiographies (3DRA) is still an open problem in the literature, with high
relevance for clinical practice. While deep learning models have been applied
for segmenting the brain vasculature in these images, they have never been
used in cases with bAVMs. This is likely caused by the difficulty to obtain
sufficiently annotated data to train these approaches. In this paper we
introduce a first deep learning model for blood vessel segmentation in 3DRA
images of patients with bAVMs. To this end, we densely annotated 5 3DRA
volumes of bAVM cases and used these to train two alternative 3DUNet-based
architectures with different segmentation objectives. Our results show that
the networks reach a comprehensive coverage of relevant structures for bAVM
analysis, much better than what is obtained using standard methods. This is
promising for achieving a better topological and morphological
characterisation of the bAVM structures of interest. Furthermore, the models
have the ability to segment venous structures even when missing in the ground
truth labelling, which is relevant for planning interventional treatments.
Ultimately, these results could be used as more reliable first initial
guesses, alleviating the cumbersome task of creating manual labels.

###### keywords:

Brain arteriovenous malformations, vessel segmentation, deep learning, 3DRA,
UNet

##  1 Introduction

Brain arteriovenous malformations (bAVMs) are an entanglement of abnormal
vessels, produced when dilated feeding arteries shunt blood directly into
arterialised draining veins[1, 2]. If untreated, these pathological
alterations have an associated risk of hemorrhages, with significant morbidity
and mortality rates[3]. Indications for bAVMs treatment vary according to its
classification—depending on factors such as the nidus size, number of feeding
arteries and draining veins (hereinafter called structures of interest) and
its anatomical location—and on whether it is ruptured or not [4, 5].

Different imaging protocols are used depending on the clinical scenario [6].
In non-traumatic settings, computed tomography angiography (CTA) or magnetic
resonance angiography (MRA) are frequently used as initial 3D examinations,
since they do not require catheterisation. For intraoperative assistance, 3D
rotational angiography (3DRA) provides high-resolution images of the vascular
anatomy of the bAVM. In this study we focus on this particular modality, which
allows for a clearer definition of the structures of interest [1].

Determining the morphology and topology of a bAVM using 3DRA allows to
efficiently extract vital information for clinicians, such as size, shape,
location and number of each of the structures of interest. The adequate
estimation of these morphological properties depends highly on the acquisition
of a precise vascular segmentation. Computational techniques can help to
alleviate the extremely tedious and time-consuming task of manually
delineating these structures. However, brain vessel segmentation remains an
open problem, specially in challenging scenarios such as bAVMs [7, 8].

As recently stated by Moccia et al. [9], the effectiveness of these methods is
determined not only by the algorithm itself but also by the imaging modality,
the presence/absence of noise or artifacts, and the anatomical region of
interest. Moreover, assessment of the segmentation quality varies according to
the task at hand. For example, in applications such as simulating the
deployment of intrasaccular devices for treating aneurysms [10, 11], it is
only required to have an accurate segmentation within the aneurysm region.
Standard intensity-based segmentation methods—such as thresholding or region
growing—achieve good results in this setting because relevant foreground
voxels share uniformity. bAVM studies, on the other hand, require a robust
coverage of low-intensity distal vessels and of the noisy entanglement of the
nidus and draining veins. While non-deep learning-based alternatives have been
proposed specifically for 3DRA images with bAVMs, these had either scarce[12]
or no[13, 14] quantitative validation on real images, or rely heavily on user
input[13], thus hampering their reproducibility.

Deep learning approaches have been used on a wide variety of biomedical
imaging problems[15, 16], including the segmentation of the brain vasculature.
However, they are either applied on healthy vasculatures[17, 18] or in images
of subjects suffering from other conditions such as acute stroke and steno-
occlusive disease[19]. To the best of our knowledge, no methods have been
specifically introduced or designed for bAVM segmentation. We believe this
could be due to the difficulties in producing an appropriate training set for
this task, as irregular blood flow in bAVMs compromises the consistency of the
contrast distribution [20]. Thus, without enough morphological references to
rely on, manual annotation becomes tedious, extremely time-consuming and prone
to errors and high inter- and intra-observer variability. On the other hand,
this also hampers the performance of standard methods to produce first
trustworthy initial guesses to work on, requiring an intense posterior
polishing labour. Furthermore, bAVMs are a rare condition, which reduces the
number of available images to annotate. As a result, collecting a large enough
comprehensive data set with densely annotated scans for training deep models
becomes prohibitive.

In this paper we propose to overcome this limitation by introducing the first
deep learning approach for segmenting challenging bAVMs on 3DRA images. To
this end, we densely annotated all brain vessel structures, including feeding
arteries, draining veins and the AVM nidus and fistulas, on a private dataset
of 5 volumetric 3DRA scans of patients with bAVMs. The resulting dataset
allowed us to perform a first comparative study of segmentation performance in
the context of bAVM analysis. By building on top of the self-adapting UNet
model introduced by Isensee et al.[21], we defined a 3D patch-based
architecture trained with a combination of a standard segmentation losses and
a soft centerline Dice[22] based objective. Despite being trained with a
reduced amount of data, our final model shows to be accurate enough to display
the regions of interest of the bAVM, consistently finding distal arteries and
venous structures that sometimes were not even present in the actual ground
truth. Hence, this early model could be used to produce first initial guesses
of the vasculature in a more trustworthy way, easing the task of generating
manual ground truth labels. Furthermore, it opens the way towards applications
such as achieving a better delineation of the nidus drainage, assisting
transvenous interventional planning and achieving a larger labelled dataset to
train more robust solutions. This line of work holds clinical relevance as it
aims towards acquiring a quantitative assessment specific to each patient,
which would ultimately allow for a better treatment outcome.

##  2 Methods and Materials

###  2.1 Dataset and image acquisition

Our dataset comprises 5 contrast-enhanced 3DRA scans of subjects with bAVM (3
assigned female at birth, with ages ranging from 23 to 55 years), collected at
the Changhai Hospital (Naval Medical University, Shanghai, P. R. China) before
interventional treatment. Data collection was approved by the clinical review
board of this medical institution. The images were acquired with a Siemens
AXIOM-Artis LEO22541 station with an isometric voxel size of 0.36mm–0.46mm,
resulting in high resolution volumes of 397 to 511 slices of $512\times 512$
pixels each.

###  2.2 Segmentation model

---
Figure 1:  Architecture of the proposed 3DUNet for brain vessel segmentation
in bAVM cases. Inputs are patches of size $64\times 64\times 64$ pixels,
extracted from the normalised 3DRA volume. Downsampling is performed with
strided convolutions and upsampling with transposed convolutions. The last
four layers of the decoder produce auxiliary segmentation outputs that are
used for deep supervision.

####  2.2.1 Network architecture

All five volumes were cropped to the region of interest, removing sections
without vasculature. Images were upsampled to have a uniform spacing of
0.227mm across all cases. Volume voxels were also normalised according to the
mean and standard deviation of the intensities of each image.

Our network is based on the nnUNet[21] framework, which allows to find optimal
configurations of UNets[23, 24] for biomedical image segmentation by self-
adapting them to heterogeneous datasets. We adapted the available source code
to train the UNet models with self-configured parameters, such as number of
layers and convolutions, as well as patch and batch size.

Due to the limited amount of samples, we trained five models to segment 3DRA
images of bAVMs through a leave-one-out cross-validation [25] configuration.
Given that our images are volumetric, using the whole angiography as input to
the network is computationally prohibitive, as the amount of RAM required to
process the produced feature maps exponentially surpassed our hardware
disponibility. Additionally, patch-based training is a useful strategy to
maximise training data. Hence, to train a 3DUnet architecture as depicted in
Figure 1, we used patches of $64\times 64\times 64$ pixels.

The encoder in our 3DUNet performs four downsampling operations with strided
convolutions, starting with 32 kernels and duplicating them at each layer. The
decoder uses transposed convolutions to upsample the input feature maps. Each
convolutional layer is followed by instance normalisation and leaky ReLU as a
non-linearity. Additional segmentation outputs are obtained from the last four
layers for deep supervision[26].

####  2.2.2 Loss function

Two different objectives were used for training, seeking the most adequate
approach to segment the vessel structures in images with bAVMs. Firstly, we
followed a combined cross-entropy and Dice based loss approach [27], namely:

$\mathcal{L}_{\text{combo}}=(1-\alpha)\mathcal{L}_{\text{CE}}+\alpha\mathcal{L}_{\text{Dice}},$

where $\mathcal{L}_{\text{CE}}$ represents the cross-entropy loss,
$\mathcal{L}_{\text{Dice}}$ the soft Dice loss, and $\alpha\in[0,1]$ the
weighted contribution of the Dice term to $\mathcal{L}_{\text{combo}}$. The
Dice loss is widely used in the context of image segmentation, particularly in
presence of a class-imbalanced problem. As Dice is undefined when the target
of interest is not present in the image, the cross-entropy loss is added as an
extra term to penalize errors in these cases and to provide gradient
stability. Furthermore, notice that, as the optimal value of Dice coefficient
is $1$ (given with absolute prediction-ground truth overlap), its value is
internally multiplied by $-1$ to optimise the Dice loss function by its
minimum value.

Additionally, we defined a second loss function which includes a centerline-
focused metric:

$\mathcal{L}_{\text{combo+clDice}}=(1-\alpha)\mathcal{L}_{\text{CE}}+\alpha((1-\beta)\mathcal{L}_{\text{Dice}}+\beta\mathcal{L}_{\text{clDice}}),$

where $\mathcal{L}_{\text{clDice}}$ is a soft centerline Dice loss
function[22] that favours overlap based on a centerline approximation and
vascular connection, instead of sparse voxel overlap. The soft centerline Dice
coefficient is internally multiplied by $-1$ to obtain its loss value.
$\beta\in[0,1]$ is the weighted contribution of $\mathcal{L}_{\text{clDice}}$
to the overlap term. We implemented this function to work on 3D images,
obtaining soft-skeletons based on pooling operations.

####  2.2.3 Experimental setup

---
Figure 2:  Schematic overview of the leave-one-out cross-validation training.
Each case in turn is reserved to act as test set. The network is trained then
over the remaining four: three cases serving as training set and the last one
as validation. The result is set of five 3DUNets trained over our five
annotated cases.

In a leave-one-out cross-validation fashion as shown in Figure 2, each of the
five models was trained using a fixed batch size of 2 $64\times 64\times 64$
pixels patches randomly extracted from a set of 4 cases (3 volumes serving as
train set and 1 as validation). Initial learning rate of 0.01, stochastic
gradient descent with Nesterov momentum ($\mu=0.99$) for 250 epochs and loss
weights $\alpha=0.5$ and $\beta=0.5$. Test case prediction was done patch-wise
using a sliding window with 50% overlap, using Gaussian-window weights[21] to
merge all score maps into a single volume and reduce stitching artifacts.
Final binarisation of the score map was done with a fixed threshold of 0.5.
The models were trained on a station with 32GB RAM, Intel Core i5-8400
processor and using a GPU NVIDIA GeForce RTX 3060 12GB.

###  2.3 Data annotation

Each study was retrospectively processed to produce reliable ground truth
annotations of the brain vasculature for both training and evaluating the
proposed model. First initial guesses of the structures of interest were
obtained by semi-automatically segmenting the major arteries using empirically
defined regions (similar to octree subdivisions) and threshold values
(adjusted individually to each region). These annotations were heavily
corrected by hand by an engineer with cerebral angiographies analysis
expertise, using the segmentation editor provided by 3D Slicer[28]. The
correction protocol consisted of adding missing branching vessels, separating
merged vessels, segmenting the AVM nidus and covering the venous drainage. The
entire manual processing took approximately 10-12 hours for each case.
Exemplary ground truth labels are shown in Figures 3 and 4.

##  3 Results

We evaluated the models both quantitatively and qualitatively, to account for
the different aspects which could be of interest considering the further study
of the obtained segmentations.

###  3.1 Quantitative results

Table 1: Quantitative results to a partial ablation study for each of the 5 cases over our two models: $\text{3DUNet}_{\text{combo}}$ trained with $\mathcal{L}_{\text{combo}}$ and $\text{3DUNet}_{\text{combo+clDice}}$ trained with $\mathcal{L}_{\text{combo+clDice}}$ , evaluated in terms of Dice, Recall (Re) and Precision (Pr) for all vessels and their centerline approximations. Case | Model | Vessel | Centerline
---|---|---|---
|  | Dice | Re | Pr | Dice | Re | Pr
Case 1 | $\text{3DUNet}_{\text{combo}}$ | 0.83 | 0.82 | 0.87 | 0.87 | 0.90 | 0.85
$\text{3DUNet}_{\text{combo+clDice}}$ | 0.83 | 0.84 | 0.82 | 0.86 | 0.91 | 0.82
Case 2 | $\text{3DUNet}_{\text{combo}}$ | 0.84 | 0.82 | 0.86 | 0.89 | 0.91 | 0.88
$\text{3DUNet}_{\text{combo+clDice}}$ | 0.84 | 0.83 | 0.85 | 0.89 | 0.88 | 0.90
Case 3 | $\text{3DUNet}_{\text{combo}}$ | 0.82 | 0.86 | 0.78 | 0.82 | 0.94 | 0.73
$\text{3DUNet}_{\text{combo+clDice}}$ | 0.80 | 0.88 | 0.74 | 0.85 | 0.95 | 0.76
Case 4 | $\text{3DUNet}_{\text{combo}}$ | 0.74 | 0.98 | 0.59 | 0.76 | 0.99 | 0.62
$\text{3DUNet}_{\text{combo+clDice}}$ | 0.72 | 0.98 | 0.57 | 0.78 | 0.99 | 0.64
Case 5 | $\text{3DUNet}_{\text{combo}}$ | 0.70 | 0.68 | 0.71 | 0.72 | 0.77 | 0.67
$\text{3DUNet}_{\text{combo+clDice}}$ | 0.70 | 0.70 | 0.70 | 0.71 | 0.77 | 0.66
Table 2: Quantitative results obtained over all our cases using thresholding, region growing and our two 3DUNet models trained with $\mathcal{L}_{\text{combo}}$ ($\text{3DUNet}_{\text{combo}}$) and $\mathcal{L}_{\text{combo+clDice}}$ ($\text{3DUNet}_{\text{combo+clDice}}$), evaluated in terms of mean $\pm$ standard deviation to Dice, Recall (Re) and Precision (Pr) for all vessels and for their centerline approximations. Method | Vessel | Centerline
---|---|---
| Dice | Re | Pr | Dice | Re | Pr
Threshold | $0.66\pm 0.17$ | $0.67\pm 0.28$ | $0.78\pm 0.14$ | $0.62\pm 0.19$ | $0.66\pm 0.33$ | $0.74\pm 0.18$
Region Growing | $0.63\pm 0.22$ | $0.52\pm 0.24$ | $\textbf{0.93}\pm 0.03$ | $0.62\pm 0.29$ | $0.52\pm 0.29$ | $\textbf{0.95}\pm 0.03$
$\text{3DUNet}_{\text{combo}}$ | $\textbf{0.78}\pm 0.05$ | $0.83\pm 0.09$ | $0.76\pm 0.09$ | $0.80\pm 0.06$ | $\textbf{0.90}\pm 0.07$ | $0.75\pm 0.10$
$\text{3DUNet}_{\text{combo+clDice}}$ | $0.77\pm 0.05$ | $\textbf{0.85}\pm 0.09$ | $0.74\pm 0.09$ | $\textbf{0.81}\pm 0.06$ | $\textbf{0.90}\pm 0.07$ | $0.76\pm 0.09$

Quantitative results for each case are shown in Table 1, obtained using the
proposed model trained with $\mathcal{L}_{\text{combo+clDice}}$
($\text{3DUNet}_{\text{combo+clDice}}$) and its counterpart trained using only
$\mathcal{L}_{\text{combo}}$ ($\text{3DUNet}_{\text{combo}}$). For this
partial ablation study, both are evaluated using the overall structures of
interest (vessels) and at centerline level in terms of Dice, Recall (Re) and
Precision (Pr). Centerline Dice values were computed based on scikit-image[29]
skeletonisations.

Average metrics over all our cases are presented in Table 2, including those
obtained with two alternative and standard segmentation methods, namely
thresholding and region growing. To favour the reproducibility of the
experiments, we used the ITK 4.13.0[30] implementations. Threshold values were
set by firstly computing the Otsu threshold and correcting it empirically to
reduce the amount of included noise. Region growing was computed by manually
selecting initial seeds inside the internal carotid artery, iterating twice
with a statistical membership criterion across an 8-connected neighbourhood
with a standard deviation multiplier of 2.

###  3.2 Qualitative results

In a general overview, our 3DUNet models reached a higher discovery rate of
the overall labelled vessel structures. In Figure 3 we present projections of
the segmentations achieved by the methods on each of the five cases, and some
relevant regions are shown enlarged in Figure 4. For most cases, thresholding
and region growing missed draining vessels and had an insufficient coverage of
venous structures present in ground truth. For example: ground truth for cases
1, 2 and 5 did not include some vein structures (like the jugular vein or
sigmoid sinus) that were segmented by the models; cases 1, 2, 3 and 5 show
that our models found the superior sagittal sinus while the standard
segmentation methods missed it, just like the fistulas in cases 2 and 5; many
distal arteries were also segmented by the models but were not present in the
ground truth, as can be observed in case 4.

---
Figure 3:  Qualitative results for each of the five cases. First column shows
a volume rendering of the corresponding case, followed by the segmentations
achieved by thresholding, region growing, $\text{3DUNet}_{\text{combo}}$ and
$\text{3DUNet}_{\text{combo+clDice}}$. Lastly, the manual annotations acting
as ground truth. Volume projections correspond to sagittal or coronal
orientations, according to the AVM location and clearest segmentation
overview.
---
Figure 4:  Enlarged view of the qualitative results presented in Figure 3. Top
to bottom: vein of case 1, fistula and venous drainage of case 2, vein
structure in case 3, distal arteries to case 4, dense nidus and draining veins
to case 5.

##  4 Discussion

We approached the segmentation of 3DRA images of bAVMs with deep learning
methods after densely annotating 5 volumes of a private dataset. Our aim was
to acquire sufficiently good binarisations that allow a posterior topological
and morphological analysis of bAVMs. The assessment was based on a
quantitative comparison of two overlap metrics: vessel Dice coefficient and
centerline Dice coefficient. These were selected to account for the overall
relevant voxels and also preservation of voxel connectivity.

We performed a partial ablation study to understand the effect of
incorporating the additional centerline-focused loss during training. No
training was carried on with only cross-entropy loss, given the class-
imbalance inherent to brain vessel images. Method comparison was narrowed down
to threshold and region growing, which are standard in vessel segmentation
tasks and their code implementations are readily available. This study design
decision was deemed appropriate to minimise reproducibility difficulties and
potential ambiguity. There are limitations to this approach, since our network
works patch-wise while threshold and region growing were used volume-wise.
However, using these standard methods on a patch level would imply threshold
level and seed computation for each patch, which would then introduce
additional complexity to the comparison.

Regarding the quantitative analysis, we considered vessel Dice and centerline
Dice coefficients with their respective Precision and Recall values. Our UNet
models consistently achieve a higher Dice coefficient value and Recall value
with respect to the standard segmentation methods, indicating a better
coverage of the foreground voxels. Even though Precision values are higher
with region growing, the resulting segmentation is insufficient in the overall
recovery of foreground voxels.

Methodologically, we observed that incorporating the additional centerline-
focused loss during training does not offer an evident improvement in the
quantitative results with respect to the combined cross-entropy and Dice
losses approach (Table 2 and Table 1). Nevertheless, qualitative analysis
hints towards clearer vessel delineations and enhanced connectivity when
employing soft centerline Dice loss calculation (as in case 4 in Figure 4).
Furthermore, considering centerline Dice as a form of quantitative report
provides additional reassurance regarding preservation of vessel-like
qualities [22].

False positives yielded by our models correspond mostly to veins and distal
arteries that were not delineated in ground truth (see Figure 3). We consider
this aspect presents two potential and interconnected advantages. Firstly,
acquiring ground truth labels for bAVMs angiographies has proven to be a
challenge, due to the lack of enough available images, high inter-rater
variability [9] and unpredictable topomorphology [1, 2]. Manually generating
these labels is an intensely time-consuming task, and even carefully polished
segmentations are prone to present missing vessels. We found that models
trained over densely annotated volumes were able to yield adequate
binarisations with high structure coverage. It would then be possible to
leverage these segmentations to ease the manual annotation task, by using the
predicted geometries as first initial guesses of the structures of interest.
Additionally, appropriately dealing with the segmentation of venous structures
is a difficult task due to shape and intensity inconsistencies. High flow
rates in bAVM cases can also trigger vascular remodeling, which further
compromises venous delineation by rendering statistical atlas unreliable.
Nonetheless, it is becoming highly relevant for clinical practice as
interventional treatments start to consider and involve transvenous
embolisation more often [31, 32, 33]. As preliminary approaches, skeleton-
based vein extractions have been proposed [34] given a sufficiently good
vascular segmentation. Hence, developing a model able to handle these
difficulties at segmentation time could be a valuable tool for embolisation
planning in clinical practice.

The presented work constitutes a further step towards achieving a robust
segmentation method despite the mentioned difficulties. This is only the first
part of the still challenging task of comprehensive technology-assisted bAVM
study, but crucial to its achievement. Our results are encouraging regarding
satisfactory coverage of all structures of interest, and might eventually help
alleviate the burdensome manual labelling task or even serve as a starting
point to reach more reliable segmentations. Ultimately, this would contribute
to a better understanding and achieving a more accurate morphological and
topological characterisation of these malformations. In this respect, further
tests are currently underway to assess the impact on model robustness of the
inclusion of diverse data augmentation techniques and synthetic vessel
datasets.

###### Acknowledgements.

CG is funded by a CONICET PhD Scholarship. This work was partially funded by
PICT startup 2021-00023 - FONCYT - ANPCYT (Argentina), PIP 2021-2023
11220200102472CO from CONICET (Argentina) and a NVIDIA Hardware Grant. It was
also partially supported by the computing facilities of Extremadura Research
Centre for Advanced Technologies (CETA-CIEMAT), funded by the European
Regional Development Fund (ERDF). CETA-CIEMAT belongs to CIEMAT and the
Government of Spain.

## References

  * [1] Dumont, A. S. et al., [Brain Arteriovenous Malformations and Arteriovenous Fistulas ], Thieme (2017).
  * [2] Rutledge, W. C. and Lawton, M. T., [Brain AVM: Current Treatments and Challenges ], ch. 4, 69–81 (2017).
  * [3] Rutledge, W. C. et al., “Hemorrhage rates and risk factors in the natural history course of brain arteriovenous malformations,” Translational stroke research 5(5), 538–542 (2014).
  * [4] Spetzler, R. F. and Martin, N. A., “A proposed grading system for arteriovenous malformations,” Journal of neurosurgery 65(4), 476–483 (1986).
  * [5] Lawton, M. T. et al., “A supplementary grading scale for selecting patients with brain arteriovenous malformations for surgery,” Neurosurgery 66(4), 702–713 (2010).
  * [6] Tranvinh, E. et al., “Contemporary imaging of cerebral arteriovenous malformations,” AJR 208(6), 1320–1330 (2017).
  * [7] Babin, D. et al., “Brain blood vessel segmentation using line-shaped profiles,” Physics in Medicine & Biology 58(22), 8041 (2013).
  * [8] Forkert, N. D. et al., “Computer-aided nidus segmentation and angiographic characterization of arteriovenous malformations,” IJCARS 8(5), 775–786 (2013).
  * [9] Moccia, S. et al., “Blood vessel segmentation algorithms—review of methods, datasets and evaluation metrics,” Computer methods and programs in biomedicine 158, 71–91 (2018).
  * [10] Dazeo, N. et al., “Intra-saccular device modeling for treatment planning of intracranial aneurysms: from morphology to hemodynamics,” International Journal of Computer Assisted Radiology and Surgery 16(10), 1663–1673 (2021).
  * [11] Muñoz, R. et al., “Simulation of intra-saccular devices for pre-operative device size selection: Method and validation for sizing and porosity simulation,” Computers in Biology and Medicine , 105744 (2022).
  * [12] Chenoune, Y. et al., “Three-dimensional segmentation and symbolic representation of cerebral vessels on 3dra images of arteriovenous malformations,” Comput. Biol. Med 115, 103489 (2019).
  * [13] Clarençon, F. et al., “Elaboration of a semi-automated algorithm for brain arteriovenous malformation segmentation: initial results,” European radiology 25(2), 436–443 (2015).
  * [14] Li, F. et al., “Segmentation and reconstruction of cerebral vessels from 3d rotational angiography for avm embolization planning,” in [2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society ], 5522–5525, IEEE (2014).
  * [15] Baldi, P., “Deep learning in biomedical data science,” Annual review of biomedical data science 1, 181–205 (2018).
  * [16] Haque, I. R. I. and Neubert, J., “Deep learning approaches to biomedical image segmentation,” Informatics in Medicine Unlocked 18, 100297 (2020).
  * [17] Phellan, R. et al., “Vascular segmentation in tof mra images of the brain using a deep convolutional neural network,” in [Intravascular Imaging and Computer Assisted Stenting, and Large-Scale Annotation of Biomedical Data and Expert Label Synthesis ], 39–46, Springer (2017).
  * [18] Chen, L. et al., “3d intracranial artery segmentation using a convolutional autoencoder,” in [2017 IEEE International Conference on Bioinformatics and Biomedicine (BIBM) ], 714–717, IEEE (2017).
  * [19] Hilbert, A. et al., “Brave-net: fully automated arterial brain vessel segmentation in patients with cerebrovascular disease,” Frontiers in artificial intelligence , 78 (2020).
  * [20] Borden, N. M., [3D angiographic atlas of neurovascular anatomy and pathology ], Cambridge University Press (2006).
  * [21] Isensee, F. et al., “nnu-net: a self-configuring method for deep learning-based biomedical image segmentation,” Nature methods 18(2), 203–211 (2021).
  * [22] Shit, S. et al., “cldice-a novel topology-preserving loss function for tubular structure segmentation,” in [Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition ], 16560–16569 (2021).
  * [23] Ronneberger, O., Fischer, P., and Brox, T., “U-net: Convolutional networks for biomedical image segmentation,” in [International Conference on Medical image computing and computer-assisted intervention ], 234–241, Springer (2015).
  * [24] Çiçek, Ö. et al., “3d u-net: learning dense volumetric segmentation from sparse annotation,” in [International conference on medical image computing and computer-assisted intervention ], 424–432, Springer (2016).
  * [25] Goodfellow, I., Bengio, Y., and Courville, A., [Deep learning ], MIT press (2016).
  * [26] Lee, C.-Y. et al., “Deeply-supervised nets,” in [Artificial intelligence and statistics ], 562–570, PMLR (2015).
  * [27] Taghanaki, S. A. et al., “Combo loss: Handling input and output imbalance in multi-organ segmentation,” Computerized Medical Imaging and Graphics 75, 24–33 (2019).
  * [28] Fedorov, A. et al., “3d slicer as an image computing platform for the quantitative imaging network,” Magnetic resonance imaging 30(9), 1323–1341 (2012).
  * [29] Van der Walt, S. et al., “scikit-image: image processing in python,” PeerJ 2, e453 (2014).
  * [30] McCormick, M. M. et al., “ITK: enabling reproducible research and open science,” Frontiers in neuroinformatics 8, 13 (2014).
  * [31] Chen, C.-J. et al., “Transvenous embolization of brain arteriovenous malformations: a review of techniques, indications, and outcomes,” Neurosurgical focus 45(1), E13 (2018).
  * [32] Chen, C.-J. et al., “Brain arteriovenous malformations: a review of natural history, pathobiology, and interventions,” Neurology 95(20), 917–927 (2020).
  * [33] Waldeck, S. et al., “First experience in the control of the venous side of the brain avm,” Journal of Clinical Medicine 10(24), 5771 (2021).
  * [34] Babin, D. et al., “Skeletonization method for vessel delineation of arteriovenous malformation,” Computers in Biology and Medicine 93, 93–105 (2018).


# Suppressing Beam Squint Effect For Near-Field Wideband Communication Through
Movable Antennas

Yanze Zhu, Qingqing Wu, Yang Liu, Qingjiang Shi, and Wen Chen  Y. Zhu, Q. Wu,
and W. Chen are with the Department of Electronic Engineering, Shanghai Jiao
Tong University, 200240, China, email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Y. Liu is with the School of
Information and Communication Engineering, Dalian University of Technology,
Dalian, China, email<EMAIL_ADDRESS>Q. Shi is with the School of
Software Engineering, Tongji University, Shanghai, China, and also with the
Shenzhen Research Institute of Big Data, Shenzhen, China, email:
<EMAIL_ADDRESS>
###### Abstract

In this correspondence, we study deploying movable antenna (MA) array in a
wideband multiple-input-single-output (MISO) communication system, where near-
field (NF) channel model is considered. To alleviate beam squint effect, we
propose to maximize the minimum analog beamforming gain across the entire
wideband spectrum by appropriately adjusting MAs’ positions, which is a highly
challenging task. By introducing a slack variable and adopting the cutting-
the-edge smoothed-gradient-descent-ascent (SGDA) method, we develop algorithms
to resolve the aforementioned challenge. Numerical results verify the
effectiveness of our proposed algorithms and demonstrate the benefit of
utilizing MA array to mitigate beam squint effect in NF wideband system.

###### Index Terms:

Movable antenna (MA), wideband communication system, near-field (NF), beam
squint effect, smoothed-gradient-descent-ascent (SGDA).

##  I Introduction

Multiple-input-multiple-output (MIMO) technology is a critical technology to
improve the spectral efficiency of the fifth-generation (5G) communication
system and future wireless communications [1]. Very recently, a novel antenna
architecture named movable antenna (MA) [2], also known as fluid antenna (FA)
[3], has been proposed to further improve the beamforming capability. As
opposed to the conventional fixed position antenna (FPA) array, which has its
antenna elements fabricated on fixed positions, MAs can move freely within a
specific area in programmable manner. This movability empowers antenna array
with additional degree-of-freedom (DoF) and hence enhances its spatial
multiplexing gain. MA array can be implemented by different techniques. For
instance, the authors of [2] proposed a structure where MAs are connected with
radio-frequency (RF)-chains and controlled by stepper motors. In [3], MA array
was constructed using liquid metal technology.

Currently, a growing body of researches are exploring the potentials of MA
array [4]-[7]. Specifically, the authors of [4] modeled the channel between a
single-MA TX and a single-MA RX and theoretically proved that employment of
MAs can improve signal-to-noise-ratio (SNR) performance. In [5], the authors
considered deploying MA in a multi-user single-input-multiple-output (MU-SIMO)
system and demonstrated that the use of MA can significantly lower the
transmit power. The authors of [6] considered maximizing the minimal signal
power of desired directions by jointly optimizing beamforming and MAs’
positions. As reported in [6], utilizing MAs can remarkably improve
beamforming gain and suppress interference. In [7], the authors investigated a
wideband single-input-single-output (SISO) system assisted by MAs and verified
that implementing MAs can boost the achievable rate based on theoretical
analysis and antenna position optimization.

Although exciting progresses have been made in the existing literature, e.g.,
[4]-[7], deploying MA array in wideband system has not yet been thoroughly
investigated, except for [7]. Note one prominent challenge to employ MIMO in
wideband system is the _beam squint effect_ [8]-[11], i.e., analog beamforming
gain degrades when subcarrier frequency deviates from the reference tone
(usually the central carrier frequency). Besides, with the enlargement of
antenna aperture, e.g., extremely-large (XL)-MIMO, and elevation of operating
frequency, e.g., millimeter-wave (mmWave) and terahertz (THz), near-field (NF)
propagation effect of electromagnetic waves will become apparent [10], [11].
However, most of the existing research, e.g., [4]-[7], are performed based on
far-field (FF) channel model. Inspired by the above observations, we are
motivated to investigate appropriate configuration method of MA array to
effectively combat beam squint effect in NF wideband communication system. Our
contributions are summarized in the following:

  * •

To the best of authors’ knowledge, this is the first work studies beam squint
effect in an NF wideband MA communication system.

  * •

To alleviate beam squint effect, we propose a max-min criterion to configure
the MAs’ positions, which maximizes the minimal analog beamforming gain over
the entire wideband spectrum.

  * •

To resolve the above challenging problem, we propose two algorithms based on
appropriate transformation and smoothed-gradient-descent-ascent (SGDA) method
[12]. Especially, the SGDA-based solution is highly efficient.

  * •

Numerical results verify the effectiveness of our proposed algorithms and
demonstrate the benefit of MA in alleviating beam squint effect in wideband
system.

Fig. 1. Wideband MISO MA communication system.

##  II System Model and Problem Formulation

As shown in Fig. 1, we consider a wideband MISO communication system where a
2-D planar MA array with $M$ antennas residing in the $yoz$-plane serves a
single-antenna user. Without loss of generality, we assume that the MA array
has a square shape of size $A\times A$, locates its bottom left vertex at the
origin of the coordinate system and arranges its neighboring edges parallel
with the $y$\- and $z$-axis, respectively. Denote
$\mathcal{M}\triangleq\\{1,\cdots,M\\}$ as the set of MAs and
$\bm{p}_{m}\triangleq[0,y_{m},z_{m}]^{\mathsf{T}},\;m\in\mathcal{M}$, as the
position of the $m$-th MA. Since all MAs should maintain within the array
surface area, the restriction
$\bm{p}_{m}\in\mathcal{C}_{\mathrm{MA}}\triangleq\\{(0,y,z)|y\in[0,A],\;z\in[0,A]\\}$,
for all $m\in\mathcal{M}$, should be satisfied with
$\mathcal{C}_{\mathrm{MA}}$ indicating the feasible dwelling region of the MAs
[4]-[7].

In this paper, we consider quasi-static NF channel model. In fact, wideband
systems generally operate in frequency band, e.g., mmWave and THz, and hence
have large Rayleigh distance. This implies that wireless channels should be
captured via more accurate NF channel model [10], [11]. Especially, in typical
NF propagation environment, line-of-sight (LoS) path significantly dominates
the non-LoS paths and therefore the latter are negligible [10], [11].
Therefore, we adopt NF LoS channel model in the following discussion. As
specified in Fig. 1, regarding the origin as the reference point, we define
$\theta_{0}$, $\phi_{0}$, and $r_{0}$ as the azimuth angle, elevation angle,
and distance of the user’s antenna, respectively.1††1Note
$(\theta_{0},\phi_{0},r_{0})$ can be estimated by various existing sensing
techniques, e.g., [13], [14]. Hence, we just assume that
$(\theta_{0},\phi_{0},r_{0})$ is known in this paper. Hence, the position of
user’s antenna is given by
$\bm{p}_{\mathrm{U}}=[r_{0}\mathsf{cos}\theta_{0}\mathsf{sin}\phi_{0},r_{0}\mathsf{sin}\theta_{0}\mathsf{sin}\phi_{0},r_{0}\mathsf{cos}\phi_{0}]^{\mathsf{T}}$.
Furthermore, given the MAs’ positions $\\{\bm{p}_{m}\\}_{m=1}^{M}$, the
normalized NF channel response vector at frequency $f$ reads

$\displaystyle\bm{b}(\\{\bm{p}_{m}\\}_{m=1}^{M},\theta_{0},\phi_{0},r_{0},f)=[$
$\displaystyle e^{-j\frac{2\pi f}{c}\|\bm{p}_{\mathrm{U}}-\bm{p}_{1}\|_{2}},$
$\displaystyle\cdots,e^{-j\frac{2\pi
f}{c}\|\bm{p}_{\mathrm{U}}-\bm{p}_{M}\|_{2}}]^{\mathsf{T}},$ (1)

where $c$ is the speed of light. Based on Fresnel approximation [15],
$\|\bm{p}_{\mathrm{U}}-\bm{p}_{m}\|_{2},\;m\in\mathcal{M}$, can be
appropriately approximated as2††2By setting $r_{0}\rightarrow\infty$, (1)
reduces to conventional FF steering vector. Hence, all discussions in this
paper can indeed be applied to FF scenario straightforwardly.

$\displaystyle\|\bm{p}_{\mathrm{U}}-\bm{p}_{m}\|_{2}\approx
r_{0}-y_{m}\mathsf{sin}\theta_{0}\mathsf{sin}\phi_{0}-z_{m}\mathsf{cos}\phi_{0}$
$\displaystyle+\frac{y_{m}^{2}+z_{m}^{2}-(y_{m}\mathsf{sin}\theta_{0}\mathsf{sin}\phi_{0}+z_{m}\mathsf{cos}\phi_{0})^{2}}{2r_{0}},\;m\in\mathcal{M}.$
(2)

Analogous to conventional hybrid beamforming architecture, whose phase-shifter
array determines the beam direction physically, MA system’s analog beam is
conquered by adjustable MAs’ positions. To investigate the beam squint effect
in wideband system, we consider the analog beamforming gain endowed by the MA
array [8]-[11]. Specifically, denote the MA array’s beam steering vector as
$\bm{w}(\\{\bm{p}_{m}\\}_{m=1}^{M},\theta_{0},\phi_{0},r_{0},f_{\mathrm{c}})$,
which is similarly expressed in (1) with $f_{\mathrm{c}}$ being the central
frequency. Then, the MAs’ analog beamforming gain is

$\mathsf{g}(\\{\bm{p}_{m}\\}_{m\\!=\\!1}^{M},f_{\mathrm{c}},f)\\!\\!\triangleq\\!\\!|\bm{w}^{\mathsf{H}}(\\{\bm{p}_{m}\\}_{m\\!=\\!1}^{M},f_{\mathrm{c}})\bm{b}(\\{\bm{p}_{m}\\}_{m\\!=\\!1}^{M},f)|,$
(3)

where $\bm{w}(\\{\bm{p}_{m}\\}_{m=1}^{M},f_{\mathrm{c}})$ and
$\bm{b}(\\{\bm{p}_{m}\\}_{m=1}^{M},f)$ respectively represent
$\bm{w}(\\{\bm{p}_{m}\\}_{m=1}^{M},\theta_{0},\phi_{0},r_{0},f_{\mathrm{c}})$
and $\bm{b}(\\{\bm{p}_{m}\\}_{m=1}^{M},\theta_{0},\phi_{0},r_{0},f)$ for
simplicity.

As indicated by (3), analog beamforming gain depends on the subcarrier
frequency $f$. For the wideband system, the carrier frequency $f$ varies over
a wide spectrum and hence incurs severe fluctuating beamforming gain across
subcarriers, which is the so-called beam squint effect. To provide more
balanced communication rate among different subcarrier’s data stream [8], [9],
we propose to maximize the minimum analog beamforming among the entire band.
Specifically, denote the subcarrier frequencies as $\\{f_{l}\\}_{l=0}^{L}$
with $f_{0}<\cdots<f_{L}$. The configuration of MAs is formulated as the
following problem

$\displaystyle(\mathcal{P}1):\max_{\\{\bm{t}_{m}\\}_{m=1}^{M}}$
$\displaystyle\min_{f\in\\{f_{l}\\}_{l=0}^{L}}\mathsf{g}^{2}(\\{\bm{t}_{m}\\}_{m=1}^{M},f_{\mathrm{c}},f)$
(4) $\displaystyle\mathrm{s.t.}\;$
$\displaystyle[0,\bm{t}_{m}^{\mathsf{T}}]^{\mathsf{T}}\in\mathcal{C}_{\mathrm{MA}},\;m\in\mathcal{M},$
(4a) $\displaystyle\|\bm{t}_{m}-\bm{t}_{s}\|_{2}\geq
D_{\mathrm{min}},\;m,s\\!\in\\!\mathcal{M},\;m\\!\neq\\!s,$ (4b)

where $\bm{t}_{m}\triangleq[y_{m},z_{m}]^{\mathsf{T}},\;m\in\mathcal{M}$,
constraint (4a) requires that all MAs should be in feasible region
$\mathcal{C}_{\mathrm{MA}}$, and constraint (4b) ensures realistic spacing
between MAs [4]-[7]. Obviously, $(\mathcal{P}1)$ is highly challenging due to
its objective and nonconvex constraint (4b). In the following, we will develop
algorithms to tackle $(\mathcal{P}1)$.

##  III Algorithm Design

In this section, via introducing a slack variable, we develop algorithm to
solve the max-min problem $(\mathcal{P}1)$. Specifically, $(\mathcal{P}1)$ can
be equivalently transformed into

$\displaystyle(\mathcal{P}2):$
$\displaystyle\max_{\\{\bm{t}_{m}\\}_{m=1}^{M},\kappa}\kappa$ (5)
$\displaystyle\mathrm{s.t.}\;$
$\displaystyle\mathsf{g}^{2}(\\{\bm{t}_{m}\\}_{m=1}^{M},f_{\mathrm{c}},f_{l})\geq\kappa,\;l=0,\cdots,L,$
(5a)
$\displaystyle[0,\bm{t}_{m}^{\mathsf{T}}]^{\mathsf{T}}\in\mathcal{C}_{\mathrm{MA}},\;m\in\mathcal{M},$
(5b) $\displaystyle\|\bm{t}_{m}-\bm{t}_{s}\|_{2}\geq
D_{\mathrm{min}},\;m,s\in\mathcal{M},\;m\neq s.$ (5c)

To obtain tractable solution, in the following, we adopt block coordinate
descent (BCD) method to alternatively optimize one MA’s position at a time.
Considering the optimization of the $m$-th MA’s position alone,
$(\mathcal{P}2)$ reduces to

$\displaystyle(\mathcal{P}3):\max_{\bm{t}_{m},\kappa}\;$ $\displaystyle\kappa$
(6) $\displaystyle\mathrm{s.t.}\;$
$\displaystyle\mathsf{g}^{2}(\bm{t}_{m},f_{\mathrm{c}},f_{l})\geq\kappa,\;l=0,\cdots,L,$
(6a)
$\displaystyle[0,\bm{t}_{m}^{\mathsf{T}}]^{\mathsf{T}}\in\mathcal{C}_{\mathrm{MA}},$
(6b) $\displaystyle\|\bm{t}_{m}-\bm{t}_{s}\|_{2}\geq
D_{\mathrm{min}},\;s\in\mathcal{M},\;s\neq m.$ (6c)

Note $(\mathcal{P}3)$ is still difficult due to the nonconvex constraints (6a)
and (6c). To tackle this issue, we leverage majorization-minimization (MM)
methodology [16] to construct concave lower-bounds for the left hand side of
(6a) and (6c), respectively. Note in the following, we will use
$\mathsf{h}(\bm{t}_{m},f_{l})\triangleq\mathsf{g}^{2}(\bm{t}_{m},f_{\mathrm{c}},f_{l}),\;m\in\mathcal{M},\;l=0,\cdots,L$,
to simplify notations.

Firstly, to handle (6a), we examine the second-order Taylor expansion of the
left hand side of (6a) and replace its Hessian matrix with a smaller one in
the sense of Loewner ordering to obtain a tight globally lower bound
surrogate. Specifically, the second-order Taylor expansion of
$\mathsf{h}(\bm{t}_{m},f_{l})$ with respect to (w.r.t.) $\bm{t}_{m}$ at the
point $\bm{t}_{m}^{(n)}$ reads

$\displaystyle\mathsf{h}(\bm{t}_{m},f_{l})\approx\mathsf{h}(\bm{t}_{m}^{(n)},f_{l})+\nabla_{\bm{t}_{m}}^{\mathsf{T}}(\mathsf{h}(\bm{t}_{m},f_{l}))|_{\bm{t}_{m}=\bm{t}_{m}^{(n)}}(\bm{t}_{m}-\bm{t}_{m}^{(n)})$
$\displaystyle+\frac{1}{2}(\bm{t}_{m}-\bm{t}_{m}^{(n)})^{\mathsf{T}}\nabla_{\bm{t}_{m}}^{2}(\mathsf{h}(\bm{t}_{m},f_{l}))|_{\bm{t}_{m}=\bm{t}_{m}^{(n)}}(\bm{t}_{m}-\bm{t}_{m}^{(n)}),$
(7)

where $\nabla_{\bm{t}_{m}}(\mathsf{h}(\bm{t}_{m},f_{l}))$ and
$\nabla_{\bm{t}_{m}}^{2}(\mathsf{h}(\bm{t}_{m},f_{l}))$ are derived in
Appendix A. Furthermore, it can be observed that the Hessian matrix
$\nabla_{\bm{t}_{m}}^{2}(\mathsf{h}(\bm{t}_{m},f_{l}))$ can be lower-bounded
by
$\min\\{\lambda_{\mathrm{min}}(\nabla_{\bm{t}_{m}}^{2}(\mathsf{h}(\bm{t}_{m},f_{l}))),0\\}\mathbf{I}_{2}$,
where $\lambda_{\mathrm{min}}(\mathbf{X})$ and $\mathbf{I}_{2}$ represent the
minimal eigenvalue of matrix $\mathbf{X}$ and the identity matrix of size
$2\times 2$, respectively. Therefore, we can obtain a global concave lower-
bound of $\mathsf{h}(\bm{t}_{m},f_{l})$ by substituting
$\min\\{\lambda_{\mathrm{min}}(\nabla_{\bm{t}_{m}}^{2}(\mathsf{h}(\bm{t}_{m},f_{l}))),0\\}\mathbf{I}_{2}$
into (7).

Next, we convexify the constraint (6c), which is much easier since the left
hand side of (6c) is convex w.r.t. $\bm{t}_{m}$. By directly utilizing the
following first-order Taylor expansion

$\|\bm{t}_{m}-\bm{t}_{s}\|_{2}\geq\frac{(\bm{t}_{m}^{(n)}-\bm{t}_{s})^{\mathsf{T}}}{\|\bm{t}_{m}^{(n)}-\bm{t}_{s}\|_{2}}(\bm{t}_{m}-\bm{t}_{s}),\;s\in\mathcal{M},\;s\neq
m,$ (8)

a concave lower-bound of $\|\bm{t}_{m}-\bm{t}_{s}\|_{2}$ has been obtained.

After surrogating the left hand side of (6a) and (6c) by the two lower-bounds
developed above, we acquire the following convex optimization problem

$\displaystyle(\mathcal{P}4):$ $\displaystyle\max_{\bm{t}_{m},\kappa}\;\kappa$
(9) $\displaystyle\mathrm{s.t.}\;$
$\displaystyle\mathsf{h}(\bm{t}_{m},f_{l}|\bm{t}_{m}^{(n)})\geq\kappa,\;l=0,\cdots,L,$
(9a)
$\displaystyle[0,\bm{t}_{m}^{\mathsf{T}}]^{\mathsf{T}}\in\mathcal{C}_{\mathrm{MA}},$
(9b)
$\displaystyle\frac{(\bm{t}_{m}^{(n)}-\bm{t}_{s})^{\mathsf{T}}}{\|\bm{t}_{m}^{(n)}-\bm{t}_{s}\|_{2}}(\bm{t}_{m}-\bm{t}_{s})\geq
D_{\mathrm{min}},\;s\in\mathcal{M},\;s\neq m,$ (9c)

which can be solved by numerical solvers, e.g., CVX, where

$\displaystyle\mathsf{h}(\bm{t}_{m},f_{l}|\bm{t}_{m}^{(n)})\\!\\!\triangleq\\!\\!\mathsf{h}(\bm{t}_{m}^{(n)},f_{l})\\!\\!+\\!\\!\nabla_{\bm{t}_{m}}^{\mathsf{T}}(\mathsf{h}(\bm{t}_{m},f_{l}))|_{\bm{t}_{m}=\bm{t}_{m}^{(n)}}(\bm{t}_{m}\\!\\!-\\!\\!\bm{t}_{m}^{(n)})$
$\displaystyle\\!+\\!\frac{\min\big{\\{}\lambda_{\mathrm{min}}\big{(}\nabla_{\bm{t}_{m}}^{2}(\mathsf{h}(\bm{t}_{m},f_{l}))|_{\bm{t}_{m}=\bm{t}_{m}^{(n)}}\big{)},0\big{\\}}}{2}\|\bm{t}_{m}\\!\\!-\\!\\!\bm{t}_{m}^{(n)}\|_{2}^{2}.$
(10)

The proposed solution to $(\mathcal{P}1)$ by introducing a slack variable is
summarized in Algorithm 1.

1: Initialize feasible $\\{\bm{t}_{m}^{(0)}\\}_{m=1}^{M}$ and $n=0$;

2: repeat

3: for $m=1$ to $M$ do

4: update $(\bm{t}_{m}^{(n+1)},\kappa^{(n+\frac{m}{M})})$ by solving
$(\mathcal{P}4)$;

5: end for

6: $n:=n+1$;

7: until convergence

Algorithm 1 Solving $(\mathcal{P}1)$ based on slack variable

##  IV SGDA-Based Efficient Solution

Note one potential defect of Alg. 1 proposed above lies in its complexity. In
fact, wideband system generally has a great number, e.g., up to hundreds or
thousands, of subcarriers. This makes $(\mathcal{P}4)$ a second-order cone
programming (SOCP) problem with a large number of constraints and hence
significantly inflates its computational complexity. In this section, we
proceed to develop an efficient algorithm based on SGDA method.

We still exploit BCD framework to optimize the antenna positions. Before
invoking SGDA algorithm, we convexify constraint (4b) by (8), and regard $f$
as a continuous parameter, i.e., $f\in[f_{0},f_{L}]$.3††3This is especially
reasonable for wideband system since the entire spectrum is finely notched in
this case. Hence, the optimization for the $m$-th MA’s position can be casted
as

$\displaystyle(\mathcal{P}5):$
$\displaystyle\max_{\bm{t}_{m}}\min_{f\in[f_{0},f_{L}]}\mathsf{h}(\bm{t}_{m},f)$
(11) $\displaystyle\mathrm{s.t.}\;$
$\displaystyle[0,\bm{t}_{m}^{\mathsf{T}}]^{\mathsf{T}}\in\mathcal{C}_{\mathrm{MA}},$
(11a)
$\displaystyle\frac{(\bm{t}_{m}^{(n)}\\!-\\!\bm{t}_{s})^{\mathsf{T}}}{\|\bm{t}_{m}^{(n)}\\!-\\!\bm{t}_{s}\|_{2}}(\bm{t}_{m}\\!-\\!\bm{t}_{s})\geq
D_{\mathrm{min}},\;s\in\mathcal{M},\;s\neq m.$ (11b)

Next, we adopt SGDA method to tackle $(\mathcal{P}5)$. Note GDA methodology
solves max-min problem via alternatively conducting gradient ascent (GA) and
gradient descent (GD) updates towards maximization and minimization,
respectively. Besides, by introducing additional smoothing terms, SGDA further
improves the convergence of the standard GDA method [12]. Specifically,
according to SGDA method [12], by introducing auxiliary variables
$\\{\bm{\alpha}_{m}\\}_{m=1}^{M}$ and defining a function

$\mathsf{k}(\bm{t}_{m},\bm{\alpha}_{m};f)\triangleq\mathsf{h}(\bm{t}_{m},f)+\frac{p_{\mathrm{S}}}{2}\|\bm{t}_{m}-\bm{\alpha}_{m}\|_{2}^{2},$
(12)

where $p_{\mathrm{S}}>0$ is a pre-defined constant, we alternatively update
$\bm{t}_{m}$ in a GA manner and $f$ and $\bm{\alpha}_{m}$ in GD manners in
each iteration. Denote $(\bm{t}_{m}^{(n)},f^{(n)},\bm{\alpha}_{m}^{(n)})$ as
the solution iterate updated in the $n$-th iteration. The detailed updates
will be elaborated in the following discussions.

### A. Update of $\bm{t}_{m}$

The variable $\bm{t}_{m}$ is updated in a GA manner, given as follows

$\bm{t}_{m}^{(n+1)}:=\bm{t}_{m}^{(n)}+\eta_{\bm{t}_{m}}^{(n)}\nabla_{\bm{t}_{m}}(\mathsf{k}(\bm{t}_{m},\bm{\alpha}_{m}^{(n)};f^{(n)}))|_{\bm{t}_{m}=\bm{t}_{m}^{(n)}},$
(13)

where $\eta_{\bm{t}_{m}}^{(n)}$ is the updating step size and can be set as a
constant [12], and
$\nabla_{\bm{t}_{m}}(\mathsf{k}(\bm{t}_{m},\bm{\alpha}_{m};f))$ is

$\nabla_{\bm{t}_{m}}(\mathsf{k}(\bm{t}_{m},\bm{\alpha}_{m};f))=\nabla_{\bm{t}_{m}}(\mathsf{h}(\bm{t}_{m},f))+p_{\mathrm{S}}(\bm{t}_{m}-\bm{\alpha}_{m}),$
(14)

where $\nabla_{\bm{t}_{m}}(\mathsf{h}(\bm{t}_{m},f))$ is given in Appendix A.

Note that the newly obtained $\bm{t}_{m}^{(n+1)}$ may not satisfy (11a) or
(11b). If so, $\bm{t}_{m}^{(n+1)}$ must be projected onto the convex domain
constituted by (11a) and (11b), which is meant to solve the following
optimization

$\displaystyle(\mathcal{P}6):$
$\displaystyle\min_{\bm{x}\in\mathbb{R}^{2}}\;\frac{1}{2}\|\bm{x}-\bm{t}_{m}^{(n+1)}\|_{2}^{2}$
(15) $\displaystyle\mathrm{s.t.}\;$
$\displaystyle[0,\bm{x}^{\mathsf{T}}]^{\mathsf{T}}\in\mathcal{C}_{\mathrm{MA}},$
(15a)
$\displaystyle\frac{(\bm{t}_{m}^{(n)}\\!-\\!\bm{t}_{s})^{\mathsf{T}}}{\|\bm{t}_{m}^{(n)}\\!-\\!\bm{t}_{s}\|_{2}}(\bm{x}\\!-\\!\bm{t}_{s})\geq
D_{\mathrm{min}},\;s\in\mathcal{M},\;s\neq m.$ (15b)

Problem $(\mathcal{P}6)$ is convex which can be numerically solved.

### B. Update of $f$

The variable $f$ is updated in a GD manner, i.e.,

$f^{(n+1)}:=f^{(n)}-\eta_{f}^{(n)}\nabla_{f}(\mathsf{k}(\bm{t}_{m}^{(n+1)},\bm{\alpha}_{m}^{(n)};f))|_{f=f^{(n)}},$
(16)

where $\eta_{f}^{(n)}$ is the updating step size and can also be set as a
constant [12]. The calculation of
$\nabla_{f}(\mathsf{k}(\bm{t}_{m},\bm{\alpha}_{m};f))$ is relegated to
Appendix B.

Obviously, there exists a possibility that $f^{(n+1)}$ falls out of the
feasible domain $[f_{0},f_{L}]$. When this occurs, we should project
$f^{(n+1)}$ onto $[f_{0},f_{L}]$, that is:

  * •

If $f^{(n+1)}<f_{0}$, $f^{(n+1)}=f_{0}$,

  * •

or if $f^{(n+1)}>f_{L}$, $f^{(n+1)}=f_{L}$.

### C. Update of $\bm{\alpha}_{m}$

The variable $\bm{\alpha}_{m}$ is updated as follows [12]

$\bm{\alpha}_{m}^{(n+1)}:=\bm{\alpha}_{m}^{(n)}+\eta_{\bm{\alpha}_{m}}^{(n)}(\bm{t}_{m}^{(n+1)}-\bm{\alpha}_{m}^{(n)}),$
(17)

where $\eta_{\bm{\alpha}_{m}}^{(n)}\in(0,1]$ is a constant.

The proposed SGDA-based algorithm to tackle $(\mathcal{P}1)$ is summarized in
Algorithm 2.

1: Initialize feasible $\\{\bm{t}_{m}^{(0)}\\}_{m=1}^{M}$ and $i=0$;

2: repeat

3: for $m=1$ to $M$ do

4: set $\bm{t}_{m}^{(i,0)}:=\bm{t}_{m}^{(i)}$,
$\bm{\alpha}_{m}^{(i,0)}:=\bm{t}_{m}^{(i,0)}$ and $n=0$;

5: randomly generate $f^{(i,0)}\in[f_{0},f_{L}]$;

6: repeat

7: update $\bm{t}_{m}^{(i,n+1)}$ by (13);

8: project $\bm{t}_{m}^{(i,n+1)}$ by solving $(\mathcal{P}6)$ (if needed);

9: update $f^{(i,n+1)}$ by (16);

110: project $f^{(i,n+1)}$ by $f^{(i,n+1)}=f_{0}$

or $f^{(i,n+1)}=f_{L}$ (if needed);

11: update $\bm{\alpha}_{m}^{(i,n+1)}$ by (17);

12: $n:=n+1$;

13: until convergence

14: set $\bm{t}_{m}^{(i+1)}:=\bm{t}_{m}^{(i,\infty)}$ and
$\bm{\alpha}_{m}^{(i+1)}:=\bm{\alpha}_{m}^{(i,\infty)}$;

15: end for

16: $i:=i+1$;

17: until convergence

Algorithm 2 Solving $(\mathcal{P}1)$ based on SGDA

##  V Simulation Results

In this section, numerical results are provided to demonstrate the
effectiveness of our proposed algorithms and the benefit of deploying MA array
in wideband communication system. Unless otherwise specified, $L=256$,
$f_{0}=30\mathrm{GHz}$, $f_{L}=50\mathrm{GHz}$,
$f_{\mathrm{c}}=\frac{f_{0}+f_{L}}{2}=40\mathrm{GHz}$,
$f_{l}=f_{0}+\frac{l}{L}(f_{L}-f_{0}),\;l=0,\cdots,L$,
$A=\frac{2\sqrt{M}c}{f_{0}}$, $D_{\mathrm{min}}=\frac{c}{2f_{0}}$,
$\theta_{0}=\frac{\pi}{6}$, $\phi_{0}=\frac{\pi}{4}$, $r_{0}=10\mathrm{m}$,
$p_{\mathrm{S}}=2$, $\eta_{\bm{\alpha}_{m}}=0.5,\;m\in\mathcal{M}$.

Fig. 2 illustrates the convergence behaviours of Alg. 1 & 2\. As suggested by
Fig. 2, by utilizing Alg. 1, the objective value of $(\mathcal{P}1)$
monotonically increases and converges in ten iterations. At the same time,
SGDA-based Alg. 2 exhibits fast improvement with slight oscillations.
Interestingly, besides its lower complexity, Alg. 2 yields a superior
converged objective value over Alg. 1.

Fig. 3 plots the array gain (defined in (3)) across different subcarriers. As
reflected by Fig. 3, via appropriately configuring the antennas’ positions, MA
can remarkably mitigate the beam squint effect compared to the conventional
FPA counterpart. Note that the analog beamforming gain yielded by Alg. 2
outperforms Alg. 1 over the entire wide frequency band.

Lastly, to examine the complexity of our proposed algorithms, we compare both
average MATLAB run-time per iteration in Fig. 4. As reflected by Fig. 4, the
run-time of Alg. 1 increases intensively when the number of subcarriers $L$
grows. In contrast, the run-time per iteration of Alg. 2 is nearly insensitive
to $L$ and maintains a much lower level than that of Alg. 1.

Fig. 2. Convergence behaviours of Alg. 1 & 2.

Fig. 3. Array gain across considered frequency band.

Fig. 4. Comparison of running time between Alg. 1 & 2.

##  VI Conclusion

In this paper, a wideband MISO communication system aided by MA array under NF
channel model is considered. To overcome beam squint effect, a minimal array
gain maximization criterion is proposed to configure MAs’ positions. We
successfully develop algorithms based on introducing a slack variable and SGDA
method. The SGDA-based algorithm has much lower complexity and is especially
suitable to systems with large bandwidth. Simulation results verify that beam
squint effect can be significantly mitigated by MA array deployment.

### A. Calculation of $\nabla_{\bm{t}_{m}}(\mathsf{h}(\bm{t}_{m},f_{l}))$ and
$\nabla_{\bm{t}_{m}}^{2}(\mathsf{h}(\bm{t}_{m},f_{l}))$

Define $C_{m,l}\triangleq\sum_{n=1,n\neq
m}^{M}e^{j\frac{2\pi(f_{\mathrm{c}}-f_{l})}{c}\mathsf{a}(\bm{t}_{n})},\;m\in\mathcal{M},\;l=0,\cdots,L$,
where $\mathsf{a}(\bm{t}_{m})\triangleq
r_{0}-y_{m}\mathsf{sin}\theta_{0}\mathsf{sin}\phi_{0}-z_{m}\mathsf{cos}\phi_{0}+\frac{y_{m}^{2}+z_{m}^{2}-(y_{m}\mathsf{sin}\theta_{0}\mathsf{sin}\phi_{0}+z_{m}\mathsf{cos}\phi_{0})^{2}}{2r_{0}},\;m\in\mathcal{M}$.
Then, after some manipulations, $\mathsf{h}(\bm{t}_{m},f_{l})$ can be
expressed as

$\displaystyle\mathsf{h}(\bm{t}_{m},f_{l})$
$\displaystyle=2|C_{m,l}|\mathsf{cos}\bigg{(}\frac{2\pi(f_{\mathrm{c}}-f_{l})}{c}\mathsf{a}(\bm{t}_{m})-\angle(C_{m,l})\bigg{)}$
$\displaystyle+|C_{m,l}|^{2}+1,\;m\in\mathcal{M},\;l=0,\cdots,L,$ (18)

where $\angle(x)$ picks up the phase of input complex scalar $x$.

The gradient $\nabla_{\bm{t}_{m}}(\mathsf{h}(\bm{t}_{m},f_{l}))$ is given by

$\nabla_{\bm{t}_{m}}(\mathsf{h}(\bm{t}_{m},f_{l}))=\bigg{[}\frac{\partial\mathsf{h}(\bm{t}_{m},f_{l})}{\partial
y_{m}},\frac{\partial\mathsf{h}(\bm{t}_{m},f_{l})}{\partial
z_{m}}\bigg{]}^{\mathsf{T}},$ (19)

where

$\displaystyle\frac{\partial\mathsf{h}(\bm{t}_{m},f_{l})}{\partial
y_{m}}=-2|C_{m,l}|\frac{2\pi(f_{\mathrm{c}}-f_{l})}{c}\frac{\partial\mathsf{a}(\bm{t}_{m})}{\partial
y_{m}}$
$\displaystyle\times\mathsf{sin}\bigg{(}\frac{2\pi(f_{\mathrm{c}}-f_{l})}{c}\mathsf{a}(\bm{t}_{m})-\angle(C_{m,l})\bigg{)},$
(20) $\displaystyle\frac{\partial\mathsf{a}(\bm{t}_{m})}{\partial
y_{m}}=\frac{y_{m}-(y_{m}\mathsf{sin}\theta_{0}\mathsf{sin}\phi_{0}+z_{m}\mathsf{cos}\phi_{0})\mathsf{sin}\theta_{0}\mathsf{sin}\phi_{0}}{r_{0}}$
$\displaystyle-\mathsf{sin}\theta_{0}\mathsf{sin}\phi_{0},$ (21)
$\displaystyle\frac{\partial\mathsf{h}(\bm{t}_{m},f_{l})}{\partial
z_{m}}=-2|C_{m,l}|\frac{2\pi(f_{\mathrm{c}}-f_{l})}{c}\frac{\partial\mathsf{a}(\bm{t}_{m})}{\partial
z_{m}}$
$\displaystyle\times\mathsf{sin}\bigg{(}\frac{2\pi(f_{\mathrm{c}}-f_{l})}{c}\mathsf{a}(\bm{t}_{m})-\angle(C_{m,l})\bigg{)},$
(22) $\displaystyle\frac{\partial\mathsf{a}(\bm{t}_{m})}{\partial
z_{m}}\\!\\!=\\!\\!\frac{z_{m}\\!\\!-\\!\\!(y_{m}\mathsf{sin}\theta_{0}\mathsf{sin}\phi_{0}\\!\\!+\\!\\!z_{m}\mathsf{cos}\phi_{0})\mathsf{cos}\phi_{0}}{r_{0}}\\!-\\!\mathsf{cos}\phi_{0}.$
(23)

The Hessian matrix $\nabla_{\bm{t}_{m}}^{2}(\mathsf{h}(\bm{t}_{m},f_{l}))$ is
given as follows

$\nabla_{\bm{t}_{m}}^{2}(\mathsf{h}(\bm{t}_{m},f_{l}))=\begin{bmatrix}\frac{\partial^{2}\mathsf{h}(\bm{t}_{m},f_{l})}{\partial
y_{m}^{2}}&\frac{\partial^{2}\mathsf{h}(\bm{t}_{m},f_{l})}{\partial
y_{m}\partial z_{m}}\\\
\frac{\partial^{2}\mathsf{h}(\bm{t}_{m},f_{l})}{\partial z_{m}\partial
y_{m}}&\frac{\partial^{2}\mathsf{h}(\bm{t}_{m},f_{l})}{\partial z_{m}^{2}}\\\
\end{bmatrix},$ (24)

where

$\displaystyle\frac{\partial^{2}\mathsf{h}(\bm{t}_{m},f_{l})}{\partial
y_{m}^{2}}=-2|C_{m,l}|\frac{2\pi(f_{\mathrm{c}}-f_{l})}{c}\frac{1-(\mathsf{sin}\theta_{0}\mathsf{sin}\phi_{0})^{2}}{r_{0}}$
$\displaystyle\\!\times\\!\mathsf{sin}\bigg{(}\frac{2\pi(f_{\mathrm{c}}\\!-\\!f_{l})}{c}\mathsf{a}(\bm{t}_{m})\\!-\\!\angle(C_{m,l})\bigg{)}\\!-\\!2|C_{m,l}|\bigg{(}\frac{2\pi(f_{\mathrm{c}}\\!-\\!f_{l})}{c}\bigg{)}^{2}$
$\displaystyle\\!\times\\!\bigg{(}\frac{\partial\mathsf{a}(\bm{t}_{m})}{\partial
y_{m}}\bigg{)}^{2}\mathsf{cos}\bigg{(}\frac{2\pi(f_{\mathrm{c}}\\!-\\!f_{l})}{c}\mathsf{a}(\bm{t}_{m})\\!-\\!\angle(C_{m,l})\bigg{)},$
(25) $\displaystyle\frac{\partial^{2}\mathsf{h}(\bm{t}_{m},f_{l})}{\partial
y_{m}\partial
z_{m}}=-2|C_{m,l}|\frac{2\pi(f_{\mathrm{c}}-f_{l})}{c}\frac{-\mathsf{sin}\theta_{0}\mathsf{sin}\phi_{0}\mathsf{cos}\phi_{0}}{r_{0}}$
$\displaystyle\\!\times\\!\mathsf{sin}\bigg{(}\frac{2\pi(f_{\mathrm{c}}\\!-\\!f_{l})}{c}\mathsf{a}(\bm{t}_{m})\\!-\\!\angle(C_{m,l})\bigg{)}\\!-\\!2|C_{m,l}|\bigg{(}\frac{2\pi(f_{\mathrm{c}}\\!-\\!f_{l})}{c}\bigg{)}^{2}$
$\displaystyle\\!\times\\!\frac{\partial\mathsf{a}(\bm{t}_{m})}{\partial
y_{m}}\frac{\partial\mathsf{a}(\bm{t}_{m})}{\partial
z_{m}}\mathsf{cos}\bigg{(}\frac{2\pi(f_{\mathrm{c}}\\!-\\!f_{l})}{c}\mathsf{a}(\bm{t}_{m})\\!-\\!\angle(C_{m,l})\bigg{)},$
(26) $\displaystyle\frac{\partial^{2}\mathsf{h}(\bm{t}_{m},f_{l})}{\partial
z_{m}\partial y_{m}}=\frac{\partial^{2}\mathsf{h}(\bm{t}_{m},f_{l})}{\partial
y_{m}\partial z_{m}},$ (27)
$\displaystyle\frac{\partial^{2}\mathsf{h}(\bm{t}_{m},f_{l})}{\partial
z_{m}^{2}}=-2|C_{m,l}|\frac{2\pi(f_{\mathrm{c}}-f_{l})}{c}\frac{1-\mathsf{cos}^{2}\phi_{0}}{r_{0}}$
$\displaystyle\\!\times\\!\mathsf{sin}\bigg{(}\frac{2\pi(f_{\mathrm{c}}\\!-\\!f_{l})}{c}\mathsf{a}(\bm{t}_{m})\\!-\\!\angle(C_{m,l})\bigg{)}\\!-\\!2|C_{m,l}|\bigg{(}\frac{2\pi(f_{\mathrm{c}}\\!-\\!f_{l})}{c}\bigg{)}^{2}$
$\displaystyle\\!\times\\!\bigg{(}\frac{\partial\mathsf{a}(\bm{t}_{m})}{\partial
z_{m}}\bigg{)}^{2}\mathsf{cos}\bigg{(}\frac{2\pi(f_{\mathrm{c}}\\!-\\!f_{l})}{c}\mathsf{a}(\bm{t}_{m})\\!-\\!\angle(C_{m,l})\bigg{)}.$
(28)

### B. Calculation of $\nabla_{f}(\mathsf{k}(\bm{t}_{m},\bm{\alpha}_{m};f))$

The gradient $\nabla_{f}(\mathsf{k}(\bm{t}_{m},\bm{\alpha}_{m};f))$ can be
written as

$\displaystyle\nabla_{f}(\mathsf{k}(\bm{t}_{m},\bm{\alpha}_{m};f))\\!\\!$
$\displaystyle=\\!\\!\bm{w}^{\mathsf{H}}\frac{\partial\bm{b}(f)}{\partial
f}\bm{b}^{\mathsf{H}}(f)\bm{w}\\!\\!+\\!\\!\bm{w}^{\mathsf{H}}\bm{b}(f)\bigg{(}\frac{\partial\bm{b}(f)}{\partial
f}\bigg{)}^{\mathsf{H}}\bm{w}$
$\displaystyle=2\mathsf{Re}\bigg{\\{}\bm{w}^{\mathsf{H}}\frac{\partial\bm{b}(f)}{\partial
f}\bm{b}^{\mathsf{H}}(f)\bm{w}\bigg{\\}},$ (29)

where $\bm{w}$ and $\bm{b}(f)$ are short for
$\bm{w}(\\{\bm{t}_{m}\\}_{m=1}^{M},f_{\mathrm{c}})$ and
$\bm{b}(\\{\bm{t}_{m}\\}_{m=1}^{M},f)$, respectively. Besides, the $m$-th
element of $\frac{\partial\bm{b}(f)}{\partial f}$ is given by

$\bigg{[}\frac{\partial\bm{b}(f)}{\partial
f}\bigg{]}_{m}=-j\frac{2\pi}{c}\mathsf{a}(\bm{t}_{m})[\bm{b}(f)]_{m},\;m\in\mathcal{M}.$
(30)

## References

  * [1] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” _IEEE Commun. Mag._ , vol. 52, no. 2, pp. 186–195, Feb. 2014.
  * [2] L. Zhu, W. Ma, and R. Zhang, “Movable antennas for wireless communication: Opportunities and challenges,” _IEEE Commun. Mag._ , vol. 62, no. 6, pp. 114–120, Jun. 2024.
  * [3] K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Fluid antenna systems,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 3, pp. 1950–1962, Mar. 2021.
  * [4] L. Zhu, W. Ma, and R. Zhang, “Modeling and performance analysis for movable antenna enabled wireless communications,” _IEEE Trans. Wireless Commun._ , vol. 23, no. 6, pp. 6234–6250, Jun. 2024.
  * [5] L. Zhu, W. Ma, B. Ning, and R. Zhang, “Movable-antenna enhanced multiuser communication via antenna position optimization,” _IEEE Trans. Wireless Commun._ , vol. 23, no. 7, pp. 7214–7229, Jul. 2024.
  * [6] W. Ma, L. Zhu, and R. Zhang, “Multi-beam forming with movable-antenna array,” _IEEE Commun. Lett._ , vol. 28, no. 3, pp. 697–701, Mar. 2024.
  * [7] L. Zhu, W. Ma, Z. Xiao, and R. Zhang, “Performance analysis and optimization for movable antenna aided wideband communications,” _arXiv preprint arXiv:2401.08974_ , 2024.
  * [8] R. W. Heath Jr., N. G.-Prelcic, S. Rangan, W. Roh, and A. M. Sayeed, “An overview of signal processing techniques for millimeter wave MIMO systems,” _IEEE J. Sel. Topics Signal Process._ , vol. 10, no. 3, pp. 436–453, Apr. 2016.
  * [9] F. Gao, B. Wang, C. Xing, J. An, and G. Y. Li, “Wideband beamforming for hybrid massive MIMO terahertz communications,” _IEEE J. Sel. Areas Commun._ , vol. 39, no. 6, pp. 1725–1740, Jun. 2021.
  * [10] M. Cui, L. Dai, Z. Wang, S. Zhou, and N. Ge, “Near-field rainbow: Wideband beam training for XL-MIMO,” _IEEE Trans. Wireless Commun._ , vol. 22, no. 6, pp. 3899–3912, Jun. 2023.
  * [11] H. Luo, F. Gao, W. Yuan, and S. Zhang, “Beam squint assisted user localization in near-field integrated sensing and communications systems,” _IEEE Trans. Wireless Commun._ , vol. 23, no. 5, pp. 4504–4517, May 2024.
  * [12] J. Zhang, P. Xiao, R. Sun, and Z.-Q. Luo, “A single-loop smoothed gradient descent-ascent algorithm for nonconvex-concave min-max problems,” in _Proc. 34th Conference on Neural Information Processing Systems (NeurIPS 2020)_ , Vancouver, Canada, 2020.
  * [13] W. Ma, L. Zhu, and R. Zhang, “Compressed sensing based channel estimation for movable antenna communications,” _IEEE Commun. Lett._ , vol. 27, no. 10, pp. 2747–2751, Oct. 2023.
  * [14] Z. Xiao, S. Cao, L. Zhu, Y. Liu, X.-G. Xia, and R. Zhang, “Channel estimation for movable antenna communication systems: A framework based on compressed sensing,” _arXiv preprint arXiv:2312.06969_ , 2023.
  * [15] K. T. Selvan and R. Janaswamy, “Fraunhofer and Fresnel distances: Unified derivation for aperture antennas,” _IEEE Antennas Propag. Mag._ , vol. 59, no. 4, pp. 12–15, Aug. 2017.
  * [16] Y. Sun, P. Babu, and D. P. Palomar, “Majorization-minimization algorithms in signal processing, communications, and machine learning,” _IEEE Trans. Signal Process._ , vol. 65, no. 3, pp. 794–816, Feb. 2017.


8: Also at University of Chinese Academy of Sciences, Beijing, China
9: Also at Institute for Theoretical and Experimental Physics named by A.I.
Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia
10: Also at Joint Institute for Nuclear Research, Dubna, Russia
11: Also at Cairo University, Cairo, Egypt
12: Also at Zewail City of Science and Technology, Zewail, Egypt
13: Also at Purdue University, West Lafayette, USA
14: Also at Université de Haute Alsace, Mulhouse, France
15: Also at Erzincan Binali Yildirim University, Erzincan, Turkey
16: Also at CERN, European Organization for Nuclear Research, Geneva,
Switzerland
17: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
18: Also at University of Hamburg, Hamburg, Germany
19: Also at Brandenburg University of Technology, Cottbus, Germany
20: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary,
Debrecen, Hungary
21: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary
22: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary, Budapest, Hungary
23: Also at IIT Bhubaneswar, Bhubaneswar, India, Bhubaneswar, India
24: Also at Institute of Physics, Bhubaneswar, India
25: Also at Shoolini University, Solan, India
26: Also at University of Visva-Bharati, Santiniketan, India
27: Also at Isfahan University of Technology, Isfahan, Iran
28: Now at INFN Sezione di Bari a, Università di Bari b, Politecnico di Bari
c, Bari, Italy
29: Also at Italian National Agency for New Technologies, Energy and
Sustainable Economic Development, Bologna, Italy
30: Also at Centro Siciliano di Fisica Nucleare e di Struttura Della Materia,
Catania, Italy
31: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy
32: Also at Riga Technical University, Riga, Latvia, Riga, Latvia
33: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia
34: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
35: Also at Warsaw University of Technology, Institute of Electronic Systems,
Warsaw, Poland
36: Also at Institute for Nuclear Research, Moscow, Russia
37: Now at National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
38: Also at Institute of Nuclear Physics of the Uzbekistan Academy of
Sciences, Tashkent, Uzbekistan
39: Also at St. Petersburg State Polytechnical University, St. Petersburg,
Russia
40: Also at University of Florida, Gainesville, USA
41: Also at Imperial College, London, United Kingdom
42: Also at P.N. Lebedev Physical Institute, Moscow, Russia
43: Also at California Institute of Technology, Pasadena, USA
44: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia
45: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia
46: Also at Università degli Studi di Siena, Siena, Italy
47: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy,
Pavia, Italy
48: Also at National and Kapodistrian University of Athens, Athens, Greece
49: Also at Universität Zürich, Zurich, Switzerland
50: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria,
Vienna, Austria
51: Also at Şırnak University, Sirnak, Turkey
52: Also at Beykent University, Istanbul, Turkey, Istanbul, Turkey
53: Also at Istanbul Aydin University, Application and Research Center for
Advanced Studies (App. & Res. Cent. for Advanced Studies), Istanbul, Turkey
54: Also at Mersin University, Mersin, Turkey
55: Also at Piri Reis University, Istanbul, Turkey
56: Also at Adiyaman University, Adiyaman, Turkey
57: Also at Ozyegin University, Istanbul, Turkey
58: Also at Izmir Institute of Technology, Izmir, Turkey
59: Also at Marmara University, Istanbul, Turkey
60: Also at Kafkas University, Kars, Turkey
61: Also at Istanbul Bilgi University, Istanbul, Turkey
62: Also at Hacettepe University, Ankara, Turkey
63: Also at School of Physics and Astronomy, University of Southampton,
Southampton, United Kingdom
64: Also at IPPP Durham University, Durham, United Kingdom
65: Also at Monash University, Faculty of Science, Clayton, Australia
66: Also at Bethel University, St. Paul, Minneapolis, USA, St. Paul, USA
67: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey
68: Also at Vilnius University, Vilnius, Lithuania
69: Also at Bingol University, Bingol, Turkey
70: Also at Georgian Technical University, Tbilisi, Georgia
71: Also at Sinop University, Sinop, Turkey
72: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey
73: Also at Texas A&M University at Qatar, Doha, Qatar
74: Also at Kyungpook National University, Daegu, Korea, Daegu, Korea
75: Also at University of Hyderabad, Hyderabad, India