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Structured OFDM Modulation for XL-MIMO System with Dual-Wideband Effects

January 2025
IEEE Transactions on Wireless Communications 24(2):1510-1525

DOI:10.1109/TWC.2024.3509914

Authors:

Wei Huang

Hefei University of Technology

Haiyang Zhang

Anhui

Caihong Kai

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Extremely large-scale multiple-input multiple-output (XL-MIMO) wideband systems may exhibit the severe delay spread, due to its spatial- and frequency-wideband (dual-wideband) effects. The typical orthogonal frequency division multiplexing (OFDM) technology have to insert a larger number of cyclic prefix (CP) to overcome the inter-symbol interference (ISI) induced by delay spread. The additional CP overhead will counteract the improvement of spectral efficiency by the large antenna array. To address the issue, this paper proposes a structured OFDM (SOFDM) modulation approach to reduce the CP overhead for wideband XL-MIMO systems with dual-wideband effects. As the ability to perform SOFDM is affected by the antenna architecture, we study the modulation technique considering different antenna structures, including fully-digital, phase shifter-based hybrid array, and dynamic metasurface antenna (DMA) architectures. Specifically, we first provide a mathematical model to represent a near-field channel with dual wideband effects. Based on the channel model, we develop the SOFDM modulation and then propose a joint spatial precoding and frequency domain equalization scheme to maximize the system spectral efficiency, where the solutions of precoding/conbining and equalization matrices are derived for the three types of antenna array architectures. Numerical simulations indicate that the proposed scheme can effectively deal with the dual-wideband effects and significantly improve the spectral efficiency with low CP overhead.

Chart of wideband XL-MIMO system with structured OFDM modulation in the three type of antenna array architectures.

…

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Structured OFDM Modulation for XL-MIMO

System with Dual-Wideband Effects

Wei Huang, Member, IEEE, Lizheng Xu, Student Member, IEEE, Haiyang Zhang, Member, IEEE,

Caihong Kai, Member, IEEE, Chunguo Li, Senior Member, IEEE, and Yongming Huang, Senior Member, IEEE

Abstract—Extremely large-scale multiple-input multiple-

output (XL-MIMO) wideband systems may exhibit the severe

delay spread, due to its spatial- and frequency-wideband

(dual-wideband) effects. The typical orthogonal frequency

division multiplexing (OFDM) technology have to insert a larger

number of cyclic preﬁx (CP) to overcome the inter-symbol

interference (ISI) induced by delay spread. The additional CP

overhead will counteract the improvement of spectral efﬁciency

by the large antenna array. To address the issue, this paper

proposes a structured OFDM (SOFDM) modulation approach

to reduce the CP overhead for wideband XL-MIMO systems

with dual-wideband effects. As the ability to perform SOFDM

is affected by the antenna architecture, we study the modulation

technique considering different antenna structures, including

fully-digital, phase shifter-based hybrid array, and dynamic

metasurface antenna (DMA) architectures. Speciﬁcally, we ﬁrst

provide a mathematical model to represent a near-ﬁeld channel

with dual wideband effects. Based on the channel model, we

develop the SOFDM modulation and then propose a joint

spatial precoding and frequency domain equalization scheme

to maximize the system spectral efﬁciency, where the solutions

of precoding/conbining and equalization matrices are derived

for the three types of antenna array architectures. Numerical

simulations indicate that the proposed scheme can effectively

deal with the dual-wideband effects and signiﬁcantly improve

the spectral efﬁciency with low CP overhead.

Index Terms—Structured OFDM, dual-wideband effects, near-

ﬁeld, dynamic metasurface array

This work was supported in part by the National Natural Science Foundation

of China under Grants 62371180, 62471173, 62225107 and 62171119, in part

by the Fundamental Research Funds for the Central Universities of China

under Grants JZ2024HGTG0311 and PA2024GDSK0114, in part by the Key

Research and Development Plan of Jiangsu Province under Grant BE2021013-

3, in part by the Natural Science Foundation on Frontier Leading Technology

Basic Research Project of Jiangsu under Grant BK20222001, in part by the

Major Key Project of Peng Cheng Laboratory (PCL). An earlier version of

this paper was presented in part at the 2023 IEEE Global Communications

Conference (Globecom) [1].

W. Huang, L. Xu and C. Kai (corresponding author) are with School

of Computing Science and Information Engineering, Hefei University of

Technology, Hefei 230601, China, and also with the Intelligent Inter-

connected Systems Laboratory of Anhui Province, Hefei University of

Technology, Hefei 230009, China (e-mail: {huangwei, chkai}@hfut.edu.cn,

xulizheng@mail.hfut.edu.cn).

H. Zhang is School of Communications and Information Engineering,

Nanjing University of Posts and Telecommunications, Nanjing 210003, China

(e-mail: haiyang.zhang@njupt.edu.cn).

C. Li is with School of Information Science and Engineering, Southeast

University, Nanjing 210096, China (e-mail: chunguoli@seu.edu.cn).

Y. Huang is with National Mobile Communications Research Laboratory,

Southeast University, Nanjing 210096, China, and also with the Pervasive

Communications Center, Purple Mountain Laboratories, Nanjing 211111,

China (e-mail: huangym@seu.edu.cn).

I. INTRODUCTION

A. Background

To satisfy the exponential increase in demand for high

data trafﬁc in the upcoming sixth-generation (6G) mobile

communication, extremely large-scale multiple-input multiple-

output (XL-MIMO) have been identiﬁed as one of the key

physical layer technology to improve the spectral and energy

efﬁciency, spatial resolution, and network coverage [2], [3].

Nevertheless, a consequence of using extremely large-scale

antenna arrays is that users may be located in the near-ﬁeld

region with high probability. In this region, the channel is

formulated as the near-ﬁeld model based on the spherical-

wavefront [4]. In recent years, tremendous efforts have been

devoted into the development of the near-ﬁeld communication

with XL-MIMO, where various physical layer transmission

schemes have been proposed to help improving the above-

mentioned key performance indicator [5]–[7].

B. Related Works and Motivations

In fact, with the increasing of the array aperture, there exists

non-negligible time delays across the array aperture for the

same data symbol [8], [9]. This phenomenon is referred to as

spatial-wideband effect, especially for the XL-MIMO systems.

Furthermore, due to the multi-path propagation, the wireless

channels are usually responsible for the frequency-wideband

effect. Therefore, XL-MIMO systems will exhibit both the

spatial selectivity and frequency selectivity, i.e., dual-wideband

effects. Due to the limited size of multi-antenna or massive

antennas array, multiple-input multiple-output (MIMO) chan-

nels are mainly affected by the frequency selectivity. In the

case, orthogonal frequency division multiplexing (OFDM) and

single carrier frequency domain equalization (SC-FDE) are the

widely utilized modulation techniques to eliminate/suppress

the inter symbol interference (ISI). OFDM is capable of trans-

forming the frequency-selective channel into a set of parallel

frequency-ﬂat channels by inserting the cyclic preﬁx (CP) [10].

Thus, MIMO-OFDM system with the capability of improving

spectral efﬁciency and combating the frequency selectivity

fading has been an active research area in the past years, which

has been also applied into the millimeter wave, Terahertz

and satellite communication systems [11]–[13]. Nevertheless,

it is well known that OFDM suffers from a high peak to

average ratio (PAPR) [14]. In comparison, SC-FDE can also

suppress/eliminate the ISI with a lower PAPR by performing

the discrete fourier transform (DFT) at receiver (Rx) [15].

This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and

content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3509914

Authorized licensed use limited to: HEFEI UNIVERSITY OF TECHNOLOGY. Downloaded on December 14,2024 at 09:19:53 UTC from IEEE Xplore. Restrictions apply.

While, its bandwidth and energy management is much more

cumbersome than that of the OFDM [16].

An alternative technique called as vector OFDM (VOFDM)

was ﬁrst proposed to reduce the CP overhead and PAPR by

balancing the OFDM and SC-FDE techniques [17]. The basic

idea of VOFDM is to split the transmit symbols into the

multiple segments (termed as vectors) with the given length.

In this sense, VOFDM can be treated as a general modulation

scheme, where OFDM and SC-FDE are just two special cases.

With regard to different design aspects, VOFDM has been

applied to spectral null channels and underwater acoustic com-

munications [18], [19]. Furthermore, the principle of VOFDM

is also extended to the time-varying channel, which is called

as orthogonal time frequency space (OTFS) modulation [20].

This modulation can transform a time-varying channel into an

almost time-invariant channel in delay-Doppler domain, where

the transmit symbols are multiplexed in the near-constant

delay-Doppler domain. Therefore, the advantage of OTFS

makes it suitable for the high-mobility scenarios [21].

So far, the existing works about the VOFDM and OTFS

were focused on the frequency or time selective channels.

In this scenario, orthogonality can be maintained among

symbol vectors (subcarrier). When the channel has the the

dual-wideband effects, the spatial selectively will destroy

the orthogonality among symbol vectors and result in the

additional ISI [19]. To guarantee orthogonality of the symbol

vectors, the XL-MIMO system has to spend more CP over-

head to combat the ISI, which may counteract the improve-

ment of spectral efﬁciency from the array gain. Therefore, a

novel structured orthogonal frequency division multiplexing

(SOFDM) modulation scheme gives rise to the possibility to

reduce the CP overhead and improve the spectral efﬁciency for

XL-MIMO system. Speciﬁcally, the key idea of SOFDM lies

in the frequency domain pre-/post-equalization at each symbol

vectors (subcarriers) and spatial domain symbol vectors-based

precoding/combining. By jointly designing pre-/post-equalizer

and precoder/combiner, we can equivalently reduce the CP

overhead and system PAPR. However, the potential of SOFDM

in facilitating XL-MIMO system with practical antenna archi-

tectures has not been thoroughly studied to date.

The ability to perform SOFDM in XL-MIMO system is

dependent on the antenna array architecture. The most ﬂexible

scheme is fully digital antenna array with the capability of

ﬂexibly controlling the beam directions, where each antenna

is connected to one radio frequency (RF) chain. With the

increasing of the number of antennas, fully digital architecture

is costly in terms of hardware implementation and energy

consumption. To overcome the issue, hybrid analog/digital

architectures have been extensively studied in massive MIMO

communication systems [22], [23]. The key idea of hybrid

architecture is to partition the signal processing into analog and

digital domains, which are realized by analog phase shifters

and baseband digital signal processing with few RF chains.

However, such architecture requires a large number of phase

shifters, which leads to relatively high energy consumption.

Recently, a emerging architecture for realizing large-scale

arrays is using the dynamic metasurface antennas (DMAs),

where each cost-effective radiating element can directly adjust

the electromagnetic wave to perform analog signal processing

[24]. Moreover, DMA array is a continuous (approximately

continuous) aperture array and can achieve the densiﬁcation

conﬁguration of radiating element in a unit surface. Thus,

DMA array can provide more larger array gain and degree

of freedom [25]. Nonetheless, different from the conventional

fully digital and phase shifter-based hybrid architectures, the

signals in DMA array undergo the different path delay when

they are propagating inside the microstrip [26]. Such phe-

nomena will aggravate the spatial selectively and lead to the

severe delay spread. Hence, the fact that the signal processing

capabilities of the aforementioned antenna architectures affect

their ability to perform the SOFDM and precoding.

C. Main Contributions

The aim of this paper is to reduce the CP overhead for

XL-MIMO system with dual wideband effects. We focus on

three types of antenna architectures, including fully digital

arrays, phase-shifters based hybrid architectures, and DMAs.

By considering the more challenging XL-MIMO channel, we

extend the VOFDM modulation into the proposed a spatial-

frequency wideband effects dual mitigation technique termed

SOFDM modulation. To our best knowlege, this paper is the

ﬁrst study on the SOFDM and precoding for the XL-MIMO

system with dual wideband effects. The main contributions of

this paper are summarized as follows:

First of all, different from the conventional near-ﬁeld chan-

nel model with frequency selectively [4], [27], we represent

a mathematical model for the near-ﬁeld wideband XL-MIMO

channel with dual wideband effects, incorporating both the

frequency-and spatial-selectively. By introducing the concept

of equiphase surface with spherical wave, the channel is

modeled as a function of limited multipath parameters, i.e.,

time delay of each propagation path, complex gain, angle of

arrival (AoA), angle of departure (AoD) and distance between

the transmitter (Tx) and Rx.

Next, to draw some insights, we consider the fully digital

architecture and develop the key idea of SOFDM for trans-

forming the dual selective channels into ISI-free composite

channels with the few CP overhead, where the composite

channels has a cyclically block-banded structure. By using the

special structure of the composite channel matrix of SOFDM

transmission, the pre/post-equalization matrices are obtained

without relying on the precoder/combiner. Furthermore, based

on the derived pre-/post-equalization matrices, the optimal

precoding and received combining matrices are derived in

closed-form by using the singular value decomposition (SVD)

and water-ﬁlling power allocation approaches.

Finally, we extend the SOFDM technique into the phase

shifters-based hybrid array and DMAs architectures, respec-

tively. Due to the coupled of analog and digital precod-

ing/conbining as well as the propagation characteristic that

signals travel inside the DMAs, we exploit the asymptotic

orthogonality of XL-MIMO channels to design the hybrid

precoding/conbining, where the digital precoding and conbin-

ing matrices are obtained by the minimum mean-square error

(MMSE) and the solutions of analog precoding and conbining

This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and

content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3509914

Authorized licensed use limited to: HEFEI UNIVERSITY OF TECHNOLOGY. Downloaded on December 14,2024 at 09:19:53 UTC from IEEE Xplore. Restrictions apply.

Scatterer

......

[,0,]

ti e e v v

id idr

00,0,0,

[,, ]

rrrr





0,0, 0

,,,

[, ,]

llxlylz

rrrr

Fig. 1. An illustration of the near-ﬁeld frequency-selective effect with UPA.

are solved by the proposed coordinate descent algorithm

(CDA). As a result, the proposed schemes can asymptotically

achieve the performance of fully-digital architecture.

D. Organization

The rest of this paper is organized as follows: Section

II presents the near-ﬁeld channel model with dual-wideband

effects, and proposes the SOFDM modualtion technique

considering three types of antenna array architectures. Sec-

tion III presents the joint pre-/post-equalization and pre-

coding/combining scheme for the fully-digital architecture

with SOFDM transmission. The proposed schemes for phase

shifter-based hybrid antennas array and DMAs architectures

in Section IV. Simulation results are provided in Section V.

Finally, we conclude the paper in Section VI.

E. Notation

In this paper, the upper and lower case bold symbols

denote matrices and vectors, respectively. We utilize (·)T,(·)∗

and (·)Hto denote the transpose, conjugate and Hermitian

transpose, respectively. diag(x)is the diagonal matrix with

the vector xon its diagonal. ⊗,⊛and ⊙stand for Kronecker,

convolution and Hadamard operations, respectively. For matrix

X, the element in the i-th row and the j-th column is denoted

by X(i, j).λ(X)denotes the largest singular value of matrix

X.|x|,∥x∥and ∥X∥2denote absolute value of complex

number x, norm-2 of vector xand spectral norm of matrix

X, respectively. Symbol (x)+denotes max(x, 0) operation.

II. SY ST EM MO DE L

In this section, we ﬁrst formulate the near-ﬁeld channel

model with dual-wideband effects. After that, we present three

types of antenna architectures and the corresponding signal

model with SOFDM modulation.

A. Near-ﬁeld Channel Model with Dual-wideband Effect

In this subsection, we focus on a point-to-point commu-

nication system with uniform planar array (UPA). In such

the system, the Tx and Rx are respectively equipped with

I=Ie×Ivand Q=qe×qvantennas, where Ieand Qe

denote the number of radiating elements along the horizontal

direction, and Ivand Qvdenote the that of radiating elements

along the vertical direction, respectively. Besides, we denote de

and dvas the radiating elements spacing along the horizontal

and vertical directions, respectively.

Scatterer

Equiphase Surface

…… ……

,,,

[, ,]

llxlylz

rrrr

[0,0,0]

r

[,0, ]

ti e e v v

id idr

Reference

Point

,,1ti l t l

 rr rr‖‖‖‖

Fig. 2. An illustration of the near-ﬁeld spatial-selective effect.

1) Frequency Wideband Effect: The frequency-wideband

effect is arisen from the multi-paths propagation. For the

XL-MIMO systems, the Rx will operate in the near-ﬁeld

region with spherical wave propagation [4], where the channel

steering vector is related to the angles and distance.

To start, we ﬁrst formulate a 3D coordinate system, where

the Tx is located at the origin of coordinates and the location

of Rx is denoted by r0= [r0,x, r0,y , r0,z ]T, as shown in Fig. 1.

Then, the locations of the lth scatter, the ith transmit antenna

and the qth received antenna are respectively written as

rl= [rl,x, rl,y , rl,z ]T,rt,i = [iede,0, ivdv]T,

rr,q = [r0,x +qede, r0,y, r0,z +qvdv]T,(1)

where ie∈ {1, . . . , Ie}and qe∈ {1, . . . , Qe}denote the

index of radiating element in the horizontal direction at

Tx and Rx, respectively. Similarly, iv∈ {1, . . . , Iv}and

qv∈ {1, . . . , Qv}denote the index of radiating element in

the vertical direction at Tx and Rx, respectively. In addition,

we deﬁne i≜(iv−1)Ie+ieas the i-th radiation element at

the Tx, and q≜(qv−1)Qe+qeas the q-th radiation element

at the Rx. Thus, the near-ﬁeld transmit and received steering

vectors are respectively expressed as

at(rt,i,rl) = e−j2π

λ∥rt,1−rl∥,·· · , e−j2π

λ∥rt,I −rl∥H,

ar(rr,q,rl) = e−j2π

λ∥rl−rr,j∥,· ·· , e−j2π

λ∥rl−rr,Q∥H.(2)

Based on (2), the near-ﬁeld time domain channel response

matrix ¯

H[τ]∈CQ×Ican be presented as

H(τ) = rIQ

l=1

αlar(rt,i,rl)aH

t(rr,q,rl)δ(τ−τl),(3)

where αlis the complex path gain of the lth scatterer. and

Ldenotes the number of scatterers. τldenote the propagation

delay of path land δ(τ−τl)stands for the impulse function

corresponding to delay τl.

2) Dual Wideband Effect: Eq. (3) models the near-ﬁeld

channel with the frequency selectivity but ignores the spatial

selectivity. In practice, due to the large array size, the max-

imum delay between different antenna elements is likely to

be comparable to or larger than the symbol interval, which

results in the unsynchronized reception. In such a case, the

This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and

content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3509914

Authorized licensed use limited to: HEFEI UNIVERSITY OF TECHNOLOGY. Downloaded on December 14,2024 at 09:19:53 UTC from IEEE Xplore. Restrictions apply.

Pre-block Processing



,1t

,1tK

Pre-equalization

at Each Subcarrier

,0t



1K

Fig. 3. Chart of wideband MIMO system with structured OFDM modulation.

antenna elements may receive different symbols at the same

time. This phenomenon is called as the spatial wideband effect,

as shown in Fig. 2. Thus, the near-ﬁeld XL-MIMO channel

exhibits the dual wideband effects. To represent the channel

with dual wideband effects, we deﬁne τl,i,q as the time delay

of the lth path between the ith transmit antenna and the qth

received antenna, which is given by

τl,i,q =τl+τi(rl) + τq(rl),(4)

where τlis the delay of path lbetween the ﬁrst transmit

and ﬁrst received antenna, τi(rl)is the delay between the

ith transmit antenna and equiphase surface corresponding to

path l,τq(rl)is the delay between the qth received antenna

and equiphase surface corresponding to path l. Since the

electromagnetic wave propagates with spherical wave in the

near-ﬁeld region, the equiphase surface is the arc surface.

Therefore, by using simple geometric operation, τi(rl)and

τq(rl)are respectively expressed as

τi(rl) = ∥rt,i −rl∥−∥rt,1−rl∥

τq(rl) = ∥rr,q −rl∥−∥rr,1−rl∥

c,(5)

where cis the speed of light. Then, considering the spatial

wideband effect, the time-delay channel is represented as

H[τ] = rIQ

LXL

l=1 αlar(rt,i,rl)aH

t(rr,q,rl)⊙∆(τ−τl)

=XL−1

l=0

Hl⊙∆(τ−τl),(6)

where ∆(τ−τl)∈RQ×Istands for the delay impulse matrix

and the (i, q)th entry of which is denoted by δ(τ−τl,i,q ).

Remark 1: It is observed from (6) that the XL-MIMO

channel suffers from both the spatial and frequency selec-

tivity and the channel delay spread is deﬁned as κ=

max

{l,i,q}={l′,i′,q′}|τl,i,q −τl′,i′,q′|, which is larger than that of

conventional MIMO channel ¯κ= max

l=l′|τl−τl′|. Thus, the

typical OFDM systems have to spend more CP to suppress the

ISI. To address the issue, based on the represented near-ﬁeld

XL-MIMO channel model with dual-wideband effects, we

propose the SOFDM modulation to reduce the CP overhead,

which will be presented in following subsection.

B. System Model with Structured OFDM in Three Type of

Antenna array Architectures

In this paper, we assume that Tx transmits Nsymbols to

Rx, denoted by s= [s0, s1,·· · , sN−1]T∈CN×1. In such

a system with SOFDM modulation, the baseband unit ﬁrst

performs pre-block processing partitioning the symbol vector

sinto block structure, where Msymbols are allocated in each

subcarrier, as shown in Fig. 3. Then, the transmitted symbol

vector sis rewritten as the matrix form S∈CM×K, which is

expressed as

S= [s0,·· · ,sk,·· · ,sK−1],(7)

with

sk= [skM , skM +1,· ·· , skM +M−1]H∈CM×1,

where Kdenotes the number of subcarrier (block) and M

represents the length of symbols modulated in each subcarrier,

i.e., N=MK. Generally, it is the interference-free channel

among the subcarriers in the OFDM systems, while, there

exists Msymbols within each subcarrier in the SOFDM.

Therefore, the pre-equalization processing is necessary at each

subcarrier to mitigate the inter-symbol interference at the Tx.

We deﬁne Ut,k ∈CM×Mas the pre-equalization matrix for

subcarrier kat the Tx so the equalized symbols at subcarrier k

is written as ¯

sk=Ut,ksk,∀k. Aiming at symbols ¯

sk,∀k, the

beam pattern of the transmit signals depends on the antenna

architecture. In this paper, we focus on three types of antenna

architecture, as shown in Fig. 4: fully-digital array, phase-

shifters based fully connected array and DMA array. We will

provide the transceiver model with SOFDM modulation for

the three type of array architectures in the follows.

1) Fully-Digital Antenna Array: In the fully-digital antenna

array, each antenna is connected to a RF chain, as illustrated in

Fig. 4(a). In fact, it is impractical for the XL-MIMO systems,

especially for the millimeter wave (mmWave) and Terahertz

(THz) frequency band, because the number of RF chains is

equal to that of antenna elements. Therefore, we consider the

fully-digital structure as a baseline scheme, obtaining some

insights for the SOFDM modulation mode in the near-ﬁeld

XL-MIMO systems.

Speciﬁcally, let wk∈CI×1denote the transmit precoding

for the kth subcarrier. Due to the block structure nature of the

transmit symbols, all symbols at each subcarrier exploits the

same precoding vector wk,∀k. Thus, deﬁne the permutation

precoding matrix at subcarrier kas

Wk=IM⊗wk∈CIM ×M,∀k . (8)

Sequentially, the transmit signal xk∈CIM×1at subcarrier

kis expressed as

xk= [xH

k,0,··· ,xH

k,m,· ·· ,xH

k,M −1]H=WkUt,ksk,(9)

By using inverse discrete fourier transform (IDFT) operation,

transmit signal in the time domain ¯x[τ]∈CI M×1is given by

¯x[τ] = ¯

0[τ],·· · ,¯

M−1[τ]H=1

KXK−1

i=0 xkej2π

Kkτ .

(10)

Thus, based on the time domain channel (6) and transmit

signal ¯

x[τ]with block structure, the time domain received

signal ¯

z[τ]∈CQM×1is given by

z[τ] = ¯

H[τ]⊛¯

x[τ] + ¯

η[τ],(11)

This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and

content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3509914

Authorized licensed use limited to: HEFEI UNIVERSITY OF TECHNOLOGY. Downloaded on December 14,2024 at 09:19:53 UTC from IEEE Xplore. Restrictions apply.

Digital

Combine Equalization

Digital

Precoder

Pre-equalization

Equalization

Digital

Combine

Digital

Precoder

IDFT&

RF Chain

Pre-equalization

DFT&

RF Chain

DFT&

RF Chain

,tk

,rk

IDFT&

RF Chain

Digital

Precoder

Pre-

equalization

Digital

Combine

DFT&

RF Chain

DFT&

RF Chain

.Equalization

,tk

,rk

IDFT&

RF Chain

IDFT&

RF Chain

DFT&

RF Chain

DFT&

RF Chain

,rk

IDFT&

RF Chain

(a) The fully-digital architecture

(b) The phase-shifters based hybrid architecture

IDFT&

RF Chain

Analog Beamforming

DMA-based Analog

Beamforming

Fig. 4. Chart of wideband XL-MIMO system with structured OFDM modulation in the three type of antenna array architectures.

where ¯

η[τ]∈CQM×1denotes the noise vector. Then, the

received signal with block structure in the frequency domain

is presented in the following proposition.

Proposition 1. In the MIMO system with SOFDM modulation,

by taking each Msymbols in the time domain into one group,

we can obtain the received signal yk∈CM×1with block

structure at subcarrier k, which is given by

yk=VH

kHkxk+VH

kηk=VH

kHkWkUH

t,ksk+VH

kηk,

(12)

where Vk=IM⊗vk∈CQM×Mdenotes the received

permutation precoding matrix with vk∈CQ×1being the

received precoding vector for subcarrier k.Hk∈CQM×IM

denotes the channel matrix with block structure of subcarrier

kand ηkis the additive Gaussian noise vector.

Proof: The proof is given in Appendix A.

Discussion: When the OFDM and the SOFDM have the

same size of DFT, the number of the transmit symbols of

SOFDM system is the Mtimes to that of OFDM system.

This means that the SOFDM system uses the same number of

CP to delivery more symbols. In other words, it equivalently

reduces the CP overhead.

It is observed that each entry of received signal vector

yk,∀kis the mixture of Msymbols. Thus, frequency equal-

ization is needed to suppress the ISI at each subcarrier. To

this end, deﬁne Ur,k ∈CM×Mas the equalization matrix

for subcarrier kat Rx. Then, the received signal rk∈CM×1

through the equalizer is written as

rk=UH

r,kyk=UH

r,kVH

kHkWkUt,ksk+UH

r,kVH

kηk.(13)

Thus, the spectral efﬁciency of subcarrier kis given by

Rk= log2IM+Γ−1

kUH

r,kVH

kHkWkUt,k

×UH

t,kWH

kHH

kVkUr,k,∀k, (14)

where Γk=σ2UH

r,kVH

kVkUt,k denotes the noise covariance

matrix at subcarrier kafter receive precoding and frequency

domain equalization.

2) Phase Shifter Based Antenna Array: Hybrid antenna

arrays combine the digital signal processing with the constant-

module constraint of analog signal. In this paper, we assume

that the Tx and Rx are respectively equipped with Ntand

NrRF chains, which are much smaller than the number of

transmit/received antennas. In the fully connected architecture,

each RF chain is connected to all antenna via a phase shifter

network, as shown in Fig. 4(b). In this case, the received signal

ykin (12) is rewritten as

yk=VH

D,kVH

RFHkWRF WD,kUH

t,ksk+VH

D,kVH

RFηk,

(15)

where WD,k ∈CMNt×Mand VD,k ∈CM Nr×Mdenote the

permutation digital precoding matrix and combining matrix

for subcarrier k, respectively. WRF ∈CM I ×MNtand VRF ∈

CMQ×M Nrstand for the permutation analog precoding matrix

and combining matrix, respectively. It is worth noting that

the WRF and VRF are block diagonal matrices. The non-

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zero elements of the matrix should satisfy the unit-modulus

constraint, which can be expressed by

|WRF(it, jt)|= 1 ∀(it, jt)∈ Ft

|VRF(ir, jr)|= 1 ∀(ir, jr)∈ Fr,

where Ft(Fr) denotes the index set of non-zero elements

of permutation analog precoder (combiner). Comparing (12)

and (15), it is observed that the received signal model of

hybird array architecture is a special case of the fully-

digital architecture derived by setting Wk=WRFWD,k and

Vk=VRFVD,k for each subcarrier. Therefore, the spectral

efﬁciency at subcarrier kfor phase shifter based hybrid archi-

tecture can be computed via (14), which is expressed as

Rps

k= log2IM+Γ−1

kUH

r,kVH

D,kVH

RFHkWRF WD,kUt,k

×UH

t,kWH

D,kWH

RFHH

kVRFVD,k Ur,k.(16)

where Γk=σ2UH

r,kVH

D,kVH

RFVRF VD,kUr,k denotes the

noise covariance matrix at subcarrier kafter received com-

bining and frequency domain equalization.

3) DMA Array: As shown in Fig. 4(c), DMA array archi-

tectures are usually cpmposed of multiple microstrips (waveg-

uides), where each microstrip contains multiple metamaterial

elements embedded onto the surface of microstrip to achieve

the reconﬁgurable antennas with low cost and power consump-

tion. Besides, the elements are the subwavelength space, which

implies that larger number of elements can be packed into a

given aperture. Unlike the application of metasurface as IRSs

with the passive (or active) reﬂective capability, the active

DMAs pack large numbers of tunably radiative metamaterials

on top of waveguides, resulting in multi-antenna transceivers

with advanced signal processing capabilities.

For the transceiver in DMA, each microstrip is fed by an

RF chain equipped at the back-end of microstrip. The signals

in the microstrip will travel a different path for each element,

which leads to the different propagation delay depending on

their location [26]. For convenience, symbols Ieand Qecan

be regarded as the number of microstrips as well as Ivand

Qvdenote the number of elements in each microstrip at the

Tx and Rx, respectively. Then, element-dependent propagation

effect is expressed as

hqe,qv=e−ρqe,qv(βqe+ȷαqe),∀qe, qv,(17)

where βqeand αqedenote the waveguide attenuation coefﬁ-

cient and the wavenumber along microstrip qe, respectively.

ρqe,qvdenotes the location of element qvin microstrip qe.

Since wavenumber αqeis a linear function of the frequency,

the propagation characteristic of microstrip can be regarded

as the frequency selectivity. Hence, the time domain impulse

response ¯

Hr[τ]∈CQ×Qat Rx is written as

Hr[τ] = XLr−1

lr=0

Hr,lrδ[τ−τlr],(18)

where ¯

Hr,lr= diag(hlr,1, . . . , hlr,Q)∈CQ×Qwith hlr,q

denoting the response coefﬁcient of the (qe, qv)th element

corresponding to tap lrat the Rx and Lris the number of the

tap of microstrip at Rx. Similarly, the time domain impulse

Ht[τ]∈CI×Iat Tx is given by

Ht[τ] = XLt−1

lt=0

Ht,ltδ[τ−τlt],(19)

where ¯

Ht,lt= diag(hlt,1, . . . , hlt,I )∈CI×Iwith hlt,i

denoting the response coefﬁcient of the (ie, iv)th element

corresponding to tap ltat the Tx and Ltis the number of the

tap of microstrip at Tx. Since, the outgoing signals propagate

inside the microstrips before being radiated, the equivalent

time domain channel with DMA array is represented as

Hdma[τ] = ¯

Hr[τ]⊛¯

H[τ]⊛¯

Ht[τ]

=XLtot−1

l=0

Hdma,l ⊙∆(τ−τl)(20)

with

Hdma,l =XLr−1

lr=0 XL−1

l=0

Hr,lr¯

Hl¯

Ht,lt−lr−l

where Ltot =Lr+Lt+L−1denotes the total number of

taps of equivalent channel ¯

Hdma[τ].

Remark 2: It is observed from (20) that the delay

spread of XL-MIMO channel with DMA is deﬁned as ¨κ=

max

{l,i,q}={l′,i′,q′}|τlt+τlr+τl,i,q −τl′,i′,q′|, which is larger than

that of XL-MIMO channel κ. The increased delay spread in

DMA channels is attributed by the additional signal propaga-

tion delays introduced by the microstrip at the Tx and Rx.

Thus, OFDM systems have to spend more CP to mitigate ISI.

Besides, the frequency response of each element in wide-

band DMA system is following the Lorentzian constraint [28],

[29]. Therefore, the frequency response of the qth element on

the kth subcarrier at Rx can be expressed as

ψr,qe,qv,k =Fr,qe,qvΩ2

ΩR

r,qe,qv2−Ω2

k−jχr,qe,qvΩk

,(21)

where χr,qe,qv,Fr,qe,qv,Ωr,qe,qv, and Ωkstand for the qual-

ity factor, conﬁgurable oscillator strength, angular resonance

frequency of the (qe, qv)th element, and analogous angular

frequency, respectively. Similarly, the frequency response of

the ith element on the kth subcarrier at Tx can be denoted as

ψt,ie,iv,k =Ft,ie,ivΩ2

(ΩR

t,ie,iv)2−Ω2

k−jχt,ie,ivΩk

,(22)

where χt,ie,iv,Ft,ie,iv, and Ωt,ie,ivdenote the quality factor,

conﬁgurable oscillator strength, and angular resonance fre-

quency of the (ie, iv)th element, respectively. For DMA array

system, the received signal ykin (12) is rewritten as

yk=VH

D,kVH

dma,kHdma,k Wdma,k WD,kUH

t,ksk

+VH

D,kVH

dma,kηk,(23)

where WD,k ∈CMIe×Mand VD,k ∈CM Qe×Mdenote the

permutation digital precoding matrix and combining matrix

for subcarrier k, respectively. Wdma,k ∈CM I ×MIeand

Vdma,k ∈CMQ×M Qestand for the permutation tunable

weights matrix of all DMA elements for subcarrier kat Tx

and Rx, respectively. Note that Wdma,k and Vdma,k are block

diagonal matrices and the non-zero elements of the matrix

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should satisfy

Wdma,k(it, jt) = ψt,i,k ∈ Wt∀(ir, jr)∈ Ft

Vdma,k(ir, jr) = ψr,q,k ∈ Wr∀(ir, jr)∈ Fr,

where Ft(Fr) denotes the index set of non-zero elements of

permutation tunable weights matrix at Tx (Rx), Wt(Wr) is

the set of all possible value for coefﬁcients ψt,i,k(ψr,q,k ),

respectively. By setting Wk=Wdma,k WD,k and Vk=

Vdma,kVD,k , the spectral efﬁciency at subcarrier kfor DMA

is given by

Rdma

k= log2IM+Γ−1

kUH

r,kVH

D,kVH

dma,kHdma,k Wdma,k

×WD,kUt,k UH

t,kWH

D,kWH

dma,kHH

dma,kVdma,k VD,k Ur,k.

(24)

where Γk=σ2UH

r,kVH

D,kVH

dma,kVdma,k VD,k Ur,k denotes

the noise covariance matrix at subcarrier kafter received

combining and frequency domain equalization.

In this work, we assume that the instantaneous CSI is avail-

able. Due to the number of antennas and spherical wave propa-

gation characteristic in XL-MIMO system, channel estimation

for becomes a challenging problem. In [6] and [30], some pilot

signalling based channel estimation schemes were developed

by leveraging the sparsity of channels in the polar-domain,

where the compress sense algorithm was used to reduce the

pilot overhead. Alternately, the implicit CSI acquisition with

beam training methods has been proposed for XL-MIMO

systems [31], [32]. Furthermore, for the robust precoding

design with statistical CSI in the XL-MIMO SOFDM system,

we leave the extension for future study.

III. PROP OS ED ALGORITHM FO R MIMO-SOFDM WITH

FUL LY-DIG ITAL ARCHITECTURE

To obtain some insights, we study the design of procoders

and equalizers for the MIMO-SOFDM system with the fully-

digital architecture in this section. The goal of this paper is

to maximize the overall spectral efﬁciency by designing the

transmit/receive precoders and frequency domain equalizers.

The optimization problem for the fully digital architecture is

formulated as

maximize

{Wk,Vk,Ur,k,Ut,k }XK−1

k=0 Rk(25a)

s.t.XK−1

k=0 Tr UH

t,kWH

kWkUt,k≤P ,

(25b)

where Pis the total transmit power budget. Note that we can

not solve problem (25) by taking the precoders and equalizers

as whole, since the precoders and equalizers are processed

in the spatial and frequency domain, respectively. Thus, we

have to design precoder and equalizer matrices, respectively.

As a result, the coupling of the optimization variables and

non-convex of objective function make problem (25) difﬁcult

to solve directly. To propose an algorithm for addressing the

issue, we take the following steps: First, by utilizing the the

characteristics of the block structure channel matrix Hk,∀k,

we can decouple the equalizers with the precoders and obtain

the optimal closed-form solutions of the frequency domain

equalizer {Ut,k,Ur,k }. Then, using the Jensen’s inequality,

we obtain the upper-bound for the objective of (25). Finally,

we derive the optimal closed-form solutions for precoder and

combiner of each subcarrier by using the SVD and water-

ﬁlling power allocation methods.

A. Design of the Frequency Domain Equalizer

In this subsection, we ﬁnd the frequency domain equalizer

matrices {Ut,k,Ur,k }. It is observed that the block structure

channel matrix Hk,∀kin (A.8) is the factor circulate matrix

[33]. To see that, we provide a lemma on the factor circulate

matrix factorization.

Lemma 1. When the composite channel Hk, deﬁned in (A.8),

is the factor circulate matrix, it can be factorized as the block

diagonal form, which is written as

Hk=GHΛH

k⊗IQˆ

Hk(ΛkG⊗II),(26)

with

Hk= diag ˜

Hk,˜

Hk+K,·· · ,˜

Hk+(M−1)K∈CQM×I M ,

where G∈CM×Mis the DFT matrix and Λk=

diag 1,·· · , e−j2π(M−1)k/N denotes the diagonal matrix

corresponding to frequency shifting e−j2πmk

N.˜

Hkdenotes the

channel of subcarrier kobtained by multiplying time domain

channel H(τ)by the DFT matrix.

Proof: The proof is given in Appendix B.

Note that in (26), it is possible to eliminate the interfer-

ence of inter-symbol block by designing equalization matrix

{Ut,k,Ur,k }, due to the decoupling of any two sub-matrices

in block channel matrix ˆ

Hk,∀k. Moreover, the DFT matrix G

and diagonal matrix Λkin (26) are independent of the channel

matrix and precoding vectors, and only related with the length

of symbols Mat each subcarrier and the total number of the

symbols. Thus, by applying the change of variables WkUt,k

= (Ut,k ⊗II)Wk, the received signal is rewritten as

rk=VH

k(ΛkGUr,k)H⊗IQˆ

Hk(ΛkGUt,k)⊗IIWksk

+UH

r,kVH

kηk.(27)

Note that ΛkGin (27) is an unitary matrix because it is

the product of DFT matrix and diagonal matrix. As a result,

the interference of inter-symbol block is eliminated at the

frequency domain by setting

r,k =Ut,k = (ΛkG)H,∀k. (28)

Then, the residual spectral efﬁciency of subcarrier kis

written as

Rk= log2IM+Γ−1

kVH

kˆ

HkWkWH

kˆ

kVk.(29)

Note that VH

kˆ

HkWkand Γ−1

kare the diagonal matrices,

owing to the block diagonal structure of channel ˆ

Hkand

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precoding matrices. In this case, the residual spectral efﬁciency

of each subcarrier is further simpliﬁed into

Rk=

M−1

m=0

log21 + 1

σ2|vH

kvk|vH

k˜

Hk+mK wk

2.(30)

Thus, with equalization matrices {Ut,k,Ur,k }obtained

in (28), the transmit and received precoders are designed

to maximize the overall spectral efﬁciency by solving the

following problem:

max

{wk,vk}XK−1

k=0 Rks.t.XK−1

k=0 ∥wk∥2≤P

M.(31)

Note that maximizing the overall spectral efﬁciency in (31)

requires a joint optimization over the transmit precoder and

received precoder. Nevertheless, the joint optimization is dif-

ﬁcult, due to the coupling of transmit and received precoders

in the objective function as well as the power constraints.

B. Design of the Transmit and Received Precoders

To solve the precoders, we ﬁrst obtain the upper-bound for

the spectral efﬁciency of each subcarrier, which is given by

(a)

≤Mlog2 1 + 1

Mσ2|vH

kvk|

M−1

m=0 vH

k˜

Hk+mK wk

(b)

≤Mlog21 + 1

σ2|vH

kvk|vH

k˜

Hk+iK wk

2,(32)

where ˜

Hk+iK = arg max

0≤m≤M∥˜

Hk+mK ∥2stands for the chan-

nel matrix corresponding to the maximum spectral norm in

set {˜

Hk,·· · ,˜

Hk+MK }with ibeing the block index corre-

sponding to maximum spectral norm. (a)is due to the Jensen’

inequality and (b)holds that the Tx transmits single-stream

data during the symbol period. Note that (32) provides the

analytical expression so as to make the original optimization

problem (31) be solved. The upper bound is is found to be

slightly loose regarding the overall spectral efﬁciency and the

solution of which is the achievability result.

Thus, the optimization problem (31) is converted to max-

imize the upper-bound of the overall spectral efﬁciency by

designing the transmit and received precoders, yielding

max

{wk,vk}

K−1

k=0

log21 + 1

σ2|vH

kvk|vH

k˜

Hk+iK wk

2(33a)

s.t.XK−1

k=0 ∥wk∥2≤P

M.(33b)

Note that the solution of optimal received precoders can be

directly given by the SVD of channel ˜

Hk+iK , since the

problem is the unconstrained optimization problem for the

received precoders. However, the solution of the optimal

transmit precoders in problem (33) is not derived by the simple

SVD of the channel, due to the total power constraints. In the

following proposition, we can obtain the optimal precoders for

optimization problem (33).

Proposition 2. Deﬁne the SVD decompositions of channel

at subcarrier kas ˜

Hk+iK =FkΣkQH

k. Then, the optimal

receive and transmit precoders are respectively expressed as

v⋆

k=Fk(1,:),w⋆

k=√pkQk(1,:),∀k, (34)

where Fk(1,:) and Qk(1,:) denote the ﬁrst column of matrix

Fand Q, respectively. It is the left and right eigenvectors

corresponding to the largest singular value, respectively. pk

is the power control factor for subcarrier kwith water-ﬁlling

power allocation strategy, i.e,

pk= max 

µ−σ2

λ2˜

Hk+iK ,0

,∀k, (35)

and µsatisﬁes

XK−1

k=0 max 

µ−σ2

λ2˜

Hk+iK ,0

=P

Using proposition 2, we are able to ﬁnd the optimal solu-

tions of the precoders. Consequently, the overall algorithm to

solve problem (25) is summarized as Algorithm 1.

Algorithm 1 Solving problem (25) w.r.t. {Ut,k ,Ur,k,vk,wk}

1: Input: Composite channel Hkand set initial Mand Nc;

2: Compute equalizer {Ut,k}and {Ur,k }via (28);

3: Get procoders {vk}and {wk}via Proposition 2;

4: Output: Equalizers and precoders {Ut,k,Ur,k ,wk,vk}.

Discussion: The proposed Algorithm 1jointly design the

frequency domain equalizers and precoders, where the opti-

mization variables are decoupled and the design of the equaliz-

ers leverages the block structure of the factor circulate channel

so the equalization complexities are reduced at each subcarrier

via the block factorization (Lemma 1). Moreover, each step in

algorithm is based on the closed-form solution such that the

implementation of the proposed structure OFDM transmission

scheme is simple and low-complexity computation.

IV. PROPOSED ALGORITHM FOR MIMO-SOFDM WITH

HYBRID ARCHITECTURE

A. Solution for Phase Shifter-based Hybrid architecture

In this subsection, we consider the classic phase shifter-

based fully-connected hybrid architecture. In such the archi-

tecture, the optimization problem for maximizing the overall

spectral efﬁciency is formulated as

max XK−1

k=0 Rps

k(36a)

s.t.XK−1

k=0 ∥WRFWD,k Ut,k∥2

F≤P , (36b)

|WRF(it, jt)|= 1 ∀(it, jt)∈ Ft(36c)

|VRF(ir, jr)|= 1 ∀(ir, jr)∈ Fr,(36d)

where the optimization variables are {WRF,VRF,WD,k ,

VD,k,Ur,k ,Ut,k},Pdenotes the maximum allowable trans-

mit power. Note that optimization problem (36) is more

challenging than problem (25), because of the additional anal-

ogy precoder (combiner) and unit-modulus constraints. Fortu-

nately, similar to the the fully-digital architecture, frequency

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domain equalizer and pre-equalizer are also independent on

the precoder and combiner in this architecture, which also can

be obtained in (28). Then, the spectral efﬁciency of subcarrier

kis written as

Rps

k= log2IM+Γ−1

kVH

D,kVH

RF ˆ

HkWRFWD,k

×WH

D,kWH

RF ˆ

kVRFVD,k .(37)

Due to the block structure nature of the transmit symbols, all

symbols at each subcarrier use the same precoding. Further,

permutation matrices WD,k and VD,k can be rewritten as

WD,k =IM⊗wD,k ,VD,k =IM⊗vD,k ∀k ,

where wD,k ∈CI×1and vD,k ∈CQ×1represent the digital

precoding and combining for k-th subcarrier, respectively.

Similarly, we have

WRF =IM⊗¯

WRF ,VRF =IM⊗¯

VRF ,

where ¯

WRF ∈CI×Ntand ¯

VRF ∈CQ×Nrdenote the analog

precoding and combining matrix, respectively. In addition,

according to lemma 1,ˆ

Hkis also expressed as a block

diagonal matrix. Thus, both VH

D,kVH

RF ˆ

HkWRFWD,k and

Γ−1

kare also the diagonal matrices. The spectral efﬁciency

for subcarrier kis further simpliﬁed into

Rps

M−1

m=0

log2



1 + vH

D,k ¯

RF ˜

Hk+mK ¯

WRFwD,k 

σ2vH

D,k ¯

RF ¯

VRFvD,k 





,

(38)

Then, the optimization problem for digital and analog precod-

ing (combining) is state as

max XK−1

k=0 Rps

k(39a)

s.t.XK−1

k=0 

¯

WRFwD,k 

2

F≤P

M,(39b)

¯

WRF(i, n)= 1 ,¯

VRF(q, n)= 1 ,(39c)

where the optimization variables are {wD,k,vD,k ,¯

WRF,

VRF}. The entries of set are with the unit-modulus constraint.

Note that optimization problem (39) is non-convex, due to the

non-convex objective function, the coupling of the precoding

vectors among the subcarriers and unit-modulus constraint. To

address the issue, we also ﬁrst derive the upper-bound for the

spectral efﬁciency of each subcarrie, which is expressed as

Rps

k≤Mlog 



1 + vH

D,k ¯

RF ˜

Hk+iK ¯

WRFwD,k 

σ2vH

D,k ¯

RF ¯

VRFvD,k 





.(40)

Generally, maximizing the spectral efﬁciency requires a joint

optimization over the precoders and combiners. This paper

uses the iterate algorithm proposed in [34] and [10], where the

transmitter is ﬁrst designed with ideal receiver and then the

receiver is designed with the given transmitter. Moreover, the

number of antennas in XL-MIMO systems tends to inﬁnity

such that VRFvD,k vH

D,kVH

RF ≈QI[35]. Therefore, the

precoders design problem at the Tx is

max

K−1

k=0

log I+1

σ2˜

Hk+iK ¯

WRFwD,k wH

D,k ¯

RF ˜

k+iK 

(41a)

s.t.XK−1

k=0 

¯

WRFwD,k 

2

F≤P

M,(41b)

¯

WRF(i, n)= 1 ,(41c)

where the optimization variables are {wD,k,¯

WRF}. The chal-

lenging in solving (41) is the coupling of the digital and

analogy precoders as well as the unit-modulus constraint. To

tackle the challenging, setting wD,k = ( ¯

RFWRF )−1

2ˆ

wD,k

with ˆ

wD,k being a dummy variable, the optimization problem

(41) is equivalently written as

max

K−1

k=0

log I+1

σ2Heﬀ,k ˆ

wD,k ˆ

D,kHH

eﬀ,k (42a)

s.t.XK−1

k=0 ˆ

D,k ˆ

wD,k2≤P

M,(42b)

where the optimization variables are {ˆ

wD,k,¯

WRF}. Matrix

Heﬀ,k ≜˜

Hk+iK ¯

WRF(¯

RFWRF )−1/2denotes the effective

channel for subcarrier k. If analogy precoders ¯

WRF is given,

digital dummy precoders are found in following proposition.

Proposition 3. Let effective channel Heﬀ,k be decomposed by

SVD approach as Heﬀ,k =Fps

kΣps

k(Qps

k)H. Then, the optimal

solution of ˆ

wD,k can be represented as

w⋆

D,k =√pkQps

k(:,1) ,∀k , (43)

where Qps

k(:,1) denote the ﬁrst column of matrix Qps

k. It is

the right singular vector corresponding to the largest singular

value. pkis the power control factor for subcarrier kwith

water-ﬁlling power allocation strategy, i.e,

pk= max µ−σ2

λ2(Heﬀ,k ),0,

and µsatisﬁes

XK−1

k=0 max µ−σ2

λ2(Heﬀ,k ),0=P

Once the optimal ˆ

w⋆

D,k is obtained, the optimal digital

precoder wD,k can be caculated as

w⋆

D,k =√pk(¯

RFWRF )−1

2Qps

k(:,1) .(44)

Next, we seek to ﬁnd the optimal analogy precoders. Since

the optimal digital precoding matrices w⋆

D,k are depend only

Hk+iK and ¯

WRF, we are able to formulate the optimization

problem for the analogy precoding as

max

WRF

K−1

k=0

log 1 + pk

σ2λ2˜

Hk+iK ¯

WRF(¯

RF ¯

WRF)−1

2

(45a)

s.t.¯

WRF(i, n)= 1 .(45b)

Due to the monotonicity of log function, (45) is simpliﬁed into

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max

WRF

K−1

k=0

λ2˜

Hk+iK ¯

WRF(¯

RF ¯

WRF)−1

2(46a)

s.t.¯

WRF(i, n)= 1 .(46b)

Then, we deﬁne A≜PK−1

k=0 ˜

k+iK ˜

Hk+iK and the objective

function in (46) is further written as

XK−1

k=0 λ2˜

Hk+iK ¯

WRF(¯

RF ¯

WRF)−1

2(47)

≤XK−1

k=0 ∥˜

Hk+iK ¯

WRF(¯

RF ¯

WRF)−1

2∥2

= Tr ¯

RF XK−1

k=0

k+iK ˜

Hk+iK ¯

WRF

n=0 ¯

RF(:, n) K−1

k=0

k+iK ˜

Hk+iK !¯

WRF(:, n)!.

Therefore, the optimization problem is written as

max

WRF XNt

n=0 ¯

RF(:, n)A¯

WRF(:, n)(48a)

s.t.¯

WRF(i, n)= 1 ,(48b)

It can be seen from (48) that the contribution of each RF chain

to objective function (48a) is independent so problem (48) can

be decomposed into Ntsub-problems, that is

max

WRF ¯

RF(:, n)A¯

WRF(:, n)s.t.¯

WRF(i, n)= 1 .

(49)

In fact, (49) is equivalent to single-stream optimal transmitter

beamforming problem with per-antenna power constraint [36].

To solve (49), we have the following proposition.

Proposition 4. Any local optimal solution of (49)satisﬁes

W⋆

RF(i, n) = (1, ζ = 0;

|ζ|,otherwise,(50)

where ζ=Pℓ=iA(i, ℓ)¯

W⋆

RF(ℓ, n).

Proof: By relaxing the unit-modulus constraint of (49)

into the inequation constraint, the solution of the relaxed

problem can be obtained with checking the Karush-Kuhn-

Tucker (KKT) condition, which is omitted for brevity.

Using Proposition 4, we utilize the iterative coordinate

descent algorithm over the elements of analog precoder to

ﬁnd the local optimal solution of (49) by alternately updating

each element with all other element ﬁxed. Since the objective

function of (49) increases in each element update step, the

convergence of the algorithm is guaranteed.

Then, we seek to design the combiners to maximize the

overall spectral efﬁciency assuming that the hybrid precoders

are already designed. For a ﬁxed analog combiner, the optimal

solution for digital combiner at subcarrier k, ∀kis obtained by

the MMSE method, i.e.,

vD,k =¯

RF ˜

Hk+iK ¯

WRFwD,k wH

D,k ¯

RF ˜

k+iK ¯

VRF

+σ2¯

RF ¯

VRF−1

RF ˜

Hk+iK ¯

WRFwD,k .(51)

For extremely large-scale antenna array, we have VH

RFVRF ≈

QI[35]. The analog combiner design problem is state as

max

VRF

K−1

k=0

log I+1

σ2Q¯

RF ˜

Hk+iK ¯

WRFwD,k

×wH

D,k ¯

RF ˜

k+iK ¯

VRF(52a)

s.t.¯

VRF(q, n)= 1 .(52b)

Further, by exploiting the Jensen’s inequality, the optimization

problem for maximizing the upper-bound of (52) is given by

max

VRF

log I+1

σ2Q¯

RFB¯

VRFs.t.¯

VRF(q, n)= 1

(53)

with

B=XK−1

k=0

Hk+iK ¯

WRFwD,k wH

D,k ¯

RF ˜

k+iK .

Note that the constraints in problem (53) are decoupled. This

also enables us to use the iterative CDA to ﬁnd the local

optimal solution of (53). Speciﬁcally, the contribution of each

element of analog combiner ¯

VRF(q, n)to objective function

in (53) is given by

log I+1

σ2Q¯

RF,¯nB¯

VRF,¯n+ log 1 + 2Ren¯

RF(q, n)

Xℓ=qCn(q, ℓ)¯

VRF(ℓ, n)o+1

σ2Qω,(54)

with

ω=Cn(q, q)+2RenXq′=q ,q′′=q

RF(q′, n)

×Cn(q′, q′′)¯

VRF(q′′ , n)o

Cn=B−B¯

VRF,¯nI+1

σ2Q¯

RF,¯nB¯

VRF,¯n−1

RF,¯nB.

where ¯

VRF,¯n∈CQ×Nr−1is the sub-matrix of ¯

VRF with

the nth column of ¯

VRF removed. Since parameters Cn

and ωare independent of element ¯

VRF(q, n), the max-

imun value of function (54) can be obtained by maximize

Re{¯

RF(q, n)Pℓ=qCn(q, ℓ)¯

VRF(ℓ, n)}. Then, the optimal

solution for each element of ¯

VRF is given by

V⋆

RF(q, n) = (1, ζps = 0;

ζps

|ζps|,otherwise,(55)

where ζps =Pℓ=qCn(q, ℓ)¯

VRF(ℓ, n). Similar to Proposi-

tion. 4, we also sequentially update each element with all

other element ﬁxed and the convergence of this algorithm to

the local optimal solution of (53) is also guaranteed because

the objective function increases in each step of algorithm.

Consequently, the overall algorithm to solve problem (25) is

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summarized in Algorithm 2.

Algorithm 2 Solving problem (36) w.r.t. {WRF ,VRF,WD,k ,

VD,k,Ur,k ,Ut,k}

1: Input: Composite channel Hkand set initial Mand N;

2: Compute equalizer {Ut,k}and {Ur,k }via (28);

3: Find ¯

WRF via Proposition 4;

4: Get {wD,k}by calculating (44);

5: Find ¯

VRF by solving the problem in (53);

6: Get {vD,k}by calculating (51);

7: Output: {WRF,VRF ,WD,k,VD,k ,Ur,k,Ut,k}.

B. Solution for DMA Architecture

In this subsection, we consider the DMA architecture, and

our goal is to maximize the spectral efﬁciency by jointly de-

signing DMA weights, precoder/combiner, and equalizer/pre-

equalizer. The optimization problem is formulated as

max XK−1

k=0 Rdma

k(56a)

s.t.XK−1

k=0 ∥WD,kUt,k ∥2

F≤P , (56b)

Wdma,k(it, jt)∈ Wt∀(it, jt)∈ Ft(56c)

Vdma,k(ir, jr)∈ Wr∀(ir, jr)∈ Fr,(56d)

where {Wdma,k,Vdma,k ,WD,k ,VD,k,Ut,k ,Ur,k}are the

optimization variables. Unlike the phase-shifter based hybrid

architecture, the DMA conﬁgurable weights rely on each

subcarrier and subject to Lorentz constraint. Since the design

of the equalizer is independent on channels, the optimal

equalizer and pre-equalizer are also given by (28), according

to Lemma 1. Then, the spectral efﬁciency of subcarrier kis

written as

Rdma

k= log2IM+Γ−1

kVH

D,kVH

dma,k ˆ

Hdma,kWdma,k

×WD,kWH

D,kWH

dma,k ˆ

dma,kVdma,k VD,k .(57)

Due to the block structure nature of the transmit symbols, all

symbols at each subcarrier use the same precoding. Further,

permutation matrices Wdma,k and Vdma,k can be rewritten as

Wdma,k =IM⊗¯

Wdma,k ,Vdma,k =IM⊗¯

Vdma,k ,

where ¯

Wdma,k ∈CI×Ieand ¯

Vdma,k ∈CQ×Qedenote the

tunable weights matrix for Tx and Rx, respectively. Moreover,

the DMA conﬁgurable weights rely on each subcarrier such

that it should satisfy [37]:

Wdma,k = Blkdiag{ψt,1,k,··· ,ψt,ie,k,· ·· ,ψt,Ie,k }

Vdma,k = Blkdiag{ψr,1,k,· ·· ,ψt,qe,k ,··· ,ψr,Qe,k},

where ψt,ie,k = [ψt,ie,1,k,· ·· , ψt,ie,Iv,k ]T∈CIv×1and

ψr,qe,k = [ψr,qe,1,k,··· , ψr,qe,Qv,k]T∈CQv×1denote the

frequency response of the ieth and qeth microstrip for Tx and

Rx. Then, the spectral efﬁciency is further simpliﬁed into

Rdma

M−1

m=0

log 



1 + 



vH

D,k ¯

dma,k ˜

Hdma,k+mK ¯

Wdma,kwD,k 



σ2



vH

D,k ¯

dma,k ¯

Vdma,kvD,k 









(58)

Then, the optimization problem is state as

max XK−1

k=0 Rdma

k(59a)

s.t.XK−1

k=0 ∥wD,k∥2≤P

M,(59b)

ψt,ie,iv,k ∈ Wt∀ie, iv, k , (59c)

ψr,qe,qv,k ∈ Wr,∀qe, qv, k, (59d)

where the optimization variables are {wD,k,vD,k ,¯

Wdma,k,

Vdma,k}. To solve the problem, we ﬁrst discard the Lorentz

constraints such that the optimal precoding combining vectors

are computed as

w⋆

k=Qdma

k(:,1) ,v⋆

k=Fdma

k(:,1) ,∀k , (60)

where Fdma

kand Qdma

kare the unitary matrices obtained from

the left and right singular vectors of ˜

Hdma,k+mK , respectively.

Then, the design of transmit precoders that maximize spectral

efﬁciency can be found by instead minimizing Euclidean

distance between w⋆

kand ¯

Wdma,kwD,k , which is written as

min ∥w⋆

k−¯

Wdma,kwD,k ∥2

Fs.t.∥wD,k∥2≤P

KM ,(61)

where the optimization variables are wD,k and ¯

Wdma,k. Thus,

we adopt an iterate algorithm to solve digital and analog

precoders, respectively. Speciﬁcally, with the given ¯

Wdma,k,

the digital precoding vector wD,k is given by

wD,k =¯

dma,k ¯

Wdma,k−1¯

dma,kw⋆

k,∀k . (62)

To optimize ¯

Wdma,k for a given wD,k, by deﬁning ψt,k =

[ψT

t,1,k,· ·· ,ψT

t,Ie,k]T∈CI×1, we reformulate (61) as

min

ψt,k ∥w⋆

k−(diag(wD,k)⊗I)ψt,k∥2.(63)

Note that the solution of problem (63) can be derived by via

the least squares approach, which is given by

ψt,k =(diag(wD,k)⊗I)H(diag(wD,k)⊗I)−1

×(diag{wD,k} ⊗ I)Hw⋆

k.(64)

Based on the obtained analog precoders in (64), we design

the DMA weights by minimizing the distance of eahc element

response from its corresponding unconstrained value. Follow-

ing the frequency selective element model (22), the resulting

optimization problem can be formulated as:

min

K−1

k=0

ie=1

iv=1 

Ft,ie,ivΩ2

(ΩR

t,ie,iv)2−Ω2

k−jΩkχt,ie,iv−ψt,ie,iv,k

(65)

where optimization variables are {Ft,ie,iv, χt,ie,iv,Ωt,ie,iv}.

In fact, this problem is difﬁcult to be solved, because the

objective function is dependent on frequency index and

the non-convex structure. To solve the problem, we again

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2 4 6 8 10 12 14 16 18 20

Fig. 5. Convergence behavior of the maximum spectral efﬁciency

-20 -15 -10 -5 0 5 10

SNR(dB)

Spectral Efficiency(bps/Hz)

Fig. 6. Spectral efﬁciency versus SNR, where the dotted line and solid line

denote the schemes without and with dual-wideband effects, respectively.

adopt an alternating optimization method. In particular, for

ﬁxed {χt,ie,iv,Ωt,ie,iv}, the non-negative element oscillation

strength values are given by [37]

Ft,ie,iv=ReK−1

k=0

Ω2

k˜

ψ∗

t,ie,iv,k

(ΩR

t,ie,iv)2−Ω2

k−jΩkχt,ie,iv+

×K−1

k=0

Ω4

((ΩR

t,ie,iv)2−Ω2

k)2−Ω2

kχ2

t,ie,iv−1

(66)

Since the unconstrained weights ψt,ie,iv,k exhibit a decreasing

magnitude, it is likely that they can be well-approximated by

the output of non-linear least squares solver [38]. Similarly, the

digital and analog conbiners at the Rx can use same algorithm

with that of the Tx.

V. NUMERICAL EVAL UATION

In this section, numerical results are provided to evaluate

the performance of the proposed scheme. In our simulation,

4 5 6 7 8 9 10 11

10-3

10-2

10-1

100

Fig. 7. CCDF versus PAPR.

-10 -5 0 5 10 15 20

Fig. 8. Spectral efﬁciency versus SNR for three types of architectures, where

the element spacing within the microstrip of DMA is set to λ/4and the

separation between microstrips is λ/2as well as the antenna spacing for full

digital and phase shifter-based array are set to λ/2.

12345678910

Fig. 9. Spectral efﬁciency versus the number of RF, where number of antennas

at the Tx and Rx for both the full digital and phase-shifter array are set to

240 and the number of element within one microstrip of DMAs is set to 24.

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we assume the carrier frequency is 72 GHz and the total

bandwidth is 1 GHz. Both the Tx and Rx are equipped with

uniform planar arrays. The number of transmitted symbols is

N= 128. For SOFDM modulation, we set the symbol length

for each subcarrier modulation to M= 2, and the number

of subcarriers to K= 64. In this simulation, we assume the

Tx is located at the coordinate origin, and the Rx is located

at coordinates c0= [0,30,0]. For the XL-MIMO channel,

we assume there are L= 10 scatterers and the complex path

gains αlare derived from the channel model described in [39].

Moreover, the spectral efﬁciency for the wideband systems in

the simulation is deﬁned as

R=Ntot −Ncp

Ntot XK−1

k=0 Rk

=Ntot −Ncp

Ntot XK−1

k=0 log2(1 + SINRk),(67)

where Ntot and Ncp denote the total transmission slot and the

length of CP, respectively. Further, we use the complementary

cumulative distribution function (CCDF) to represent the

PAPR performance, which is deﬁned as

P(PAPR > η)=1−P(PAPR ≤η)=1−(1 −e−η2)K,

(68)

where Pdenotes the probability function and ηstands for the

threshold on the PAPR.

Fig. 5shows the convergence behavior of our proposed

iteration approaches for the maximum spectral efﬁciency

optimization, where the iteration procedure is performed in

solving the analog precoding operation. It is observed that the

proposed algorithm leads to monotonically increasing spectral

efﬁciency values over iterations, thus leads to a converged

solution, where the proposed algorithm is suitable for the

OFDM and SOFDM systems and the performance of SOFDM

is better than that of the OFDM.

In Fig. 6, we depict the spectrum efﬁciency versus SNR

in the different parameter settings. It is observed that the

performance of the proposed SOFDM modulation scheme is

better than that of the typical OFDM scheme, due to the

equalization and precoding strategy of SOFDM proposed in

Section III. Speciﬁcally, we are able to obtain the intra-block

diversity and multiplexing gain by partitioning the symbols

into the block structure and frequency domain equalization.

Further, the designed precoding at the transceiver can op-

timally allocate the power in the symbols and subcarriers.

Moreover, the performances of all schemes without dual-

wideband effects are better than that with dual-wideband

effects, because the systems have to spend more CP overhead.

Thus, based on (67), the spectral efﬁciency decreases as the

length of CP grows. Also, the performance is improved as the

number of antennas at Tx and Rx, this is since the systems

can achieve more greater design of the degree of freedom with

the increasing of antennas.

Fig. 7shows the CCDF versus PAPR for the different

modulation types, where the number of the transmit and

received antennas are set to 512. As expected, the performance

of SOFDM modulation is the trade-off between the OFDM

and the SC-FDE, since the number of subcarrier in SOFDM

systems is less than that of OFDM but more than SC-FDE.

Moreover, we observe that the PAPR will gradually decrease,

as the length of symbols Mgrows. An intuitive explanation

is that the system utilizes the fewer subcarriers to delivery

the same amount of symbols, according to (68). An intuitive

explanation is that the system utilizes the fewer subcarriers to

delivery the same amount of symbols.

In Fig. 8, we plot the spectral efﬁciency versus SNR for

three types of architectures, where all arrays have the same

aperture with 0.1 m and the number of RF is set to 10. The

performance of DMA architecture is close to full digital array

and better than that of phase shifter-based array architecture,

since the DMA is the continuous (approximately continuous)

aperture array and can achieve the densiﬁcation conﬁguration

of radiating element in a unit surface. Furthermore, the spectral

efﬁciency can be improved as the length of CP decreases. This

is because the spectrum resource saved by reducing the CP can

be used to data transmission stage. Besides, the performance

of full-connected structure is also better than that of partial-

connected (sub-array) structure, since the each RF makes full

use of all phase-shifters in the full-connected structure.

Fig. 9reports the spectral efﬁciency versus the number of

RF, where the transmit power is set as 15 dB. It indicates

that for all schemes except fully digital array architecture, the

performance in general improves as more RF chains, but only

marginal improvement as the number of RF chains reaches to

5and 10 for the phase shifters-based hybrid array and DMA

architectures, respectively. This is expected since performance

will be constrained by the number of paths of channels and size

of array aperture. Furthermore, the performance of DMA is

better than that of phase shifters architecture when the number

of RF exceeds 6. The reason is that the DMA is a continuous

(approximately continuous) aperture array and the number of

radiating element of DMA is larger than that of the phase

shifters-based hybrid array, as the number of RF grows.

VI. CONCLUSION

This paper proposed the structure OFDM modulation

scheme for wideband XL-MIMO systems, where a joint spatial

precoding and frequency domain equalization scheme was

developed. Speciﬁcally, we ﬁrst represented the transceiver

model for the XL-MIMO with SOFDM modulation for the

three different types of antenna architectures, including fully-

digital, phase shifters-based hybrid and DMA architectures.

Based on the model, we formulated a SOFDM modulation

scheme for maximizing the system spectral efﬁciency. After

that, we proposed the efﬁcient solution of the transmit and

received precoders as well as pre/equalizers for each of the

antenna array architectures. Numerical results demonstrated

that the proposed SOFDM modulation scheme can improve

the spectral efﬁciency and reduce the PAPR.

APPENDIX

A. Proof of Proposition 1

To perform block processing, we use the circular convolu-

tion instead of the linear convolution (11). To this end, we ﬁrst

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convert the time domain channel ¯

H[τ]into composite channel

matrix ¨

H∈CQ×IN via the zero padding operation, i.e.,

H=¯

H0⊙∆(τ−τ0)

| {z }

,·· · ,¯

Hq⊙∆(τ−τq)

| {z }

·· · ,¯

HN−1⊙∆(τ−τN−1),(A.1)

where block matrix ¨

Hq∈CQ×Iis the null matrix for q≥

L−1. Also, we rewrite ¯

x[τ]as the composite form:

x= [¯

x0[0],·· · ,¯

xM−1[0],·· · ,¯

xM−1[K−1]]

=¨x0,· · · ¨xI−1

| {z }

,·· · ,¨x(N−1)I,· · · ¨xNI −1

| {z }

xN−1∈CIN ×1.

(A.2)

Thus, the received signal is the circular convolution of the

composite signal ¨

xand channel ¨

H, which is given by

z=





zN−1





=¨

H⊛¨

x=



PN−1

q=0 ¨

Hq¨

x0−q

PN−1

q=0 ¨

Hq¨

xN−1−q





.(A.3)

To perform the block operation, we have to transform the

circular convolution (A.3) into the structure circular convolu-

tion, which is given by

Z=





z0·· · ¯

z(K−1)M)

zM−1·· · ¯

zKM −1







=XN−1

q=0 





Hq¨

x0−q·· · ¨

Hq¨

x(K−1)M−q

Hq¨

xM−1−q·· · ¨

Hq¨

xKM −q





.(A.4)

To represent the received signal of the each symbol at any

block, we deﬁne n=bM +aand q=b′M+a′such that the

received signal of the ath symbol at the bth block is given by

zbM+a=XM−1

a′=0 XK−1

b′=0

Hb′M+a′¨

xbM+a−b′M−a′

=XM−1

a′=0 XK−1

b′=0

Hb′M+a′¨

x(b−b′+ξ)M+(a−a′),(A.5)

with

ξ=(0, a −a′≥0

−1, a −a′<0

By performing K-points DFT operation, the received signal

(A.5) is transformed into the frequency domain form:

zk,a =XK−1

b=0 ¯

zbM+ae−j2π

Kbk

K−1

b=0 M−1

a′=0

K−1

b′=0

Hb′M+a′¨

x(b−b′+ξ)M+(a−a′)!e−j2π

Kbk

=XM−1

a′=0 XK−1

b′=0

Hb′M+a′xk,a−a′e−j2π

Kb′kej2π

Kkξ

=XM−1

a′=0

HkM +a′xk,a−a′ej2π

Kkξ .(A.6)

Thus, the received signal zk∈CQM×1at subcarrier kis

wiritten as

zk=Hkxk+ηk,(A.7)

with

Hk=







HkM +0 e−j2πk

K¨

HkM +M−1··· e−j2πk

K¨

HkM +1

HkM +1 ¨

HkM +0 ··· e−j2πk

K¨

HkM +2

.···

HkM +M−1¨

HkM +M−2··· ¨

HkM +0







(A.8)

At last, the baseband signal for subcarrier kvia received block

precoding matrix Vkis written as

yk=VH

kzk=VH

kHkxk+VH

kηk.(A.9)

B. Proof of Lemma 1

For the factor circulant matrix, it is the sum of Kronecker

product between the basic factor circulant matrix Θkand

block matrix ¨

HKM +m,∀m, which is expressed as

Hk=XM−1

m=0 Θm

k⊗¨

HkM +m.(B.1)

where Θm

kdenotes the Θkto the power of m. Moreover, the

basic factor circulant matrix Θkcan be decomposed into [33]

Θm

k= (ΛkG)Hˆ

Θm

kΛkG.(B.2)

where G∈CM×Mis the DFT matrix and Λk=

diag 1,·· · , e−j2π(M−1)k/N denotes the diagonal matrix.

Θm

k= diag e−j2π km/N,· ·· , e−j2π((M−1)K+k)m/N . Sub-

stituting (B.2) into (B.1), channel at subcarrier kis given by

Hk=GHΛH

k⊗INrXM−1

m=0

Θm

k⊗¨

HkM +m(ΛkG⊗INt)

=GHΛH

k⊗INrˆ

Hk(ΛkG⊗INt),(B.3)

with

Hk= diag ˜

Hk,˜

Hk+K,·· · ,˜

Hk+(M−1)K∈CQM×I M .

Here, the submatrix on the diagonal in ˆ

Hkis given by

Hk=XL−1

l=0

Hle−j2π

Nkl ∈CQ×I, k = 0,1,·· · , N −1.

REFERENCES

[1] W. Huang, L. Xu, H. Zhang, and C. Kai, “Structured OFDM design for

massive MIMO systems with dual-wideband effects,” in GLOBECOM

2023-IEEE Global Communications Conference, 2023, pp. 4050–4055.

[2] S. Hu, F. Rusek, and O. Edfors, “Beyond massive MIMO: The potential

of data transmission with large intelligent surfaces,” IEEE Trans. Signal

Process., vol. 66, no. 10, pp. 2746–2758, 2018.

[3] E. D. Carvalho, A. Ali, A. Amiri, M. Angjelichinoski, and R. W. Heath,

“Non-stationarities in extra-large-scale massive MIMO,” IEEE Wireless

Commun., vol. 27, no. 4, pp. 74–80, 2020.

[4] J. Sherman, “Properties of focused apertures in the fresnel region,” IRE

Trans. Antennas and Propag., vol. 10, no. 4, pp. 399–408, 1962.

[5] Y. Lu and L. Dai, “Near-ﬁeld channel estimation in mixed LoS/NLoS

environments for extremely large-scale MIMO systems,” IEEE Trans.

Commun., vol. 71, no. 6, pp. 3694–3707, 2023.

[6] M. Cui and L. Dai, “Channel estimation for extremely large-scale

MIMO: Far-ﬁeld or near-ﬁeld?” IEEE Trans. Commun., vol. 70, no. 4,

pp. 2663–2677, 2022.

[7] H. Zhang, N. Shlezinger, F. Guidi, D. Dardari, M. F. Imani, and Y. C.

Eldar, “Beam focusing for near-ﬁeld multiuser MIMO communications,”

IEEE Trans. Wireless Commun., vol. 21, no. 9, pp. 7476–7490, 2022.

[8] B. Wang, F. Gao, S. Jin, H. Lin, and G. Y. Li, “Spatial- and frequency-

wideband effects in millimeter-wave massive MIMO systems,” IEEE

Trans. Signal Process., vol. 66, no. 13, pp. 3393–3406, 2018.

This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and

content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3509914

Authorized licensed use limited to: HEFEI UNIVERSITY OF TECHNOLOGY. Downloaded on December 14,2024 at 09:19:53 UTC from IEEE Xplore. Restrictions apply.

[9] T.-H. Chou, N. Michelusi, D. J. Love, and J. V. Krogmeier, “Compressed

training for dual-wideband time-varying sub-terahertz massive MIMO,”

IEEE Trans. Commun., vol. 71, no. 6, pp. 3559–3575, 2023.

[10] F. Sohrabi and W. Yu, “Hybrid analog and digital beamforming for

mmwave OFDM large-scale antenna arrays,” IEEE J. Sel. Areas in

Commun., vol. 35, no. 7, pp. 1432–1443, 2017.

[11] K. M. Attiah, F. Sohrabi, and W. Yu, “Deep learning for channel sensing

and hybrid precoding in TDD massive MIMO OFDM systems,” IEEE

Trans. Wireless Commun., vol. 21, no. 12, pp. 10 839–10 853, 2022.

[12] N. T. Nguyen, N. Shlezinger, Y. C. Eldar, and M. Juntti, “Multiuser

MIMO wideband joint communications and sensing system with sub-

carrier allocation,” IEEE Trans. Signal Process., vol. 71, pp. 2997–3013,

2023.

[13] K.-X. Li, X. Gao, and X.-G. Xia, “Channel estimation for LEO satellite

massive MIMO OFDM communications,” IEEE Trans. Wireless Com-

mun., pp. 1–1, 2023.

[14] Y. Rahmatallah and S. Mohan, “Peak-to-average power ratio reduction

in OFDM systems: A survey and taxonomy,” IEEE Commun. Surveys

& Tuts., vol. 15, no. 4, pp. 1567–1592, 2013.

[15] M. Luzio, R. Dinis, and P. Montezuma, “Pragmatic frequency domain

equalization for single carrier with offset modulations,” IEEE Trans.

Wireless Commun., vol. 12, no. 9, pp. 4496–4505, 2013.

[16] C.-W. Jen, H.-C. Liu, Z.-C. Huang, N.-C. Xue, and S.-J. J. Jou, “A

digital frequency-dependent I/Q imbalance and group delay estima-

tion and compensation modules for mmwave single carrier baseband

transceivers,” IEEE J. Emerging and S. Circuits and Sys., vol. 13, no. 2,

pp. 461–475, 2023.

[17] X.-g. Xia, “Precoded and vector OFDM robust to channel spectral

nulls and with reduced cyclic preﬁx length in single transmit antenna

systems,” IEEE Trans. Commun., vol. 49, no. 8, pp. 1363–1374, 2001.

[18] H. Zhang, X.-g. Xia, Q. Zhang, and W. Zhu, “Precoded OFDM with

adaptive vector channel allocation for scalable video transmission over

frequency-selective fading channels,” IEEE Trans. Mobile Comput.,

vol. 1, no. 2, pp. 132–142, 2002.

[19] J. Han, S. P. Chepuri, Q. Zhang, and G. Leus, “Iterative per-vector equal-

ization for orthogonal signal-division multiplexing over time-varying

underwater acoustic channels,” IEEE J. Oceanic Engineer., vol. 44, no. 1,

pp. 240–255, 2019.

[20] R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F.

Molisch, and R. Calderbank, “Orthogonal time frequency space modu-

lation,” in 2017 IEEE Wireless Communications and Networking Con-

ference (WCNC), 2017, pp. 1–6.

[21] J. Shi, Z. Li, J. Hu, Z. Tie, S. Li, W. Liang, and Z. Ding, “OTFS enabled

LEO satellite communications: A promising solution to severe doppler

effects,” IEEE Network, pp. 1–7, 2023.

[22] R. W. Heath, N. Gonz´

alez-Prelcic, S. Rangan, W. Roh, and A. M.

Sayeed, “An overview of signal processing techniques for millimeter

wave MIMO systems,” IEEE J. S. Signal Process., vol. 10, no. 3, pp.

436–453, 2016.

[23] A. Pizzo, A. Lozano, S. Rangan, and T. L. Marzetta, “Wide-aperture

MIMO via reﬂection off a smooth surface,” IEEE Trans. Wireless

Commun., vol. 22, no. 8, pp. 5229–5239, 2023.

[24] N. Shlezinger, G. C. Alexandropoulos, M. F. Imani, Y. C. Eldar, and

D. R. Smith, “Dynamic metasurface antennas for 6g extreme massive

MIMO communications,” IEEE Wireless Commun., vol. 28, no. 2, pp.

106–113, 2021.

[25] I. Yoo, M. F. Imani, T. Sleasman, H. D. Pﬁster, and D. R. Smith, “En-

hancing capacity of spatial multiplexing systems using reconﬁgurable

cavity-backed metasurface antennas in clustered MIMO channels,” IEEE

Trans Commun., vol. 67, no. 2, pp. 1070–1084, 2019.

[26] N. Shlezinger, O. Dicker, Y. C. Eldar, I. Yoo, M. F. Imani, and

D. R. Smith, “Dynamic metasurface antennas for uplink massive MIMO

systems,” IEEE Trans. Commun., vol. 67, no. 10, pp. 6829–6843, 2019.

[27] M. Cui and L. Dai, “Near-ﬁeld channel estimation for extremely large-

scale MIMO with hybrid precoding,” in 2021 IEEE Global Communi-

cations Conference (GLOBECOM), 2021, pp. 1–6.

[28] H. Wang, N. Shlezinger, Y. C. Eldar, S. Jin, M. F. Imani, I. Yoo,

and D. R. Smith, “Dynamic metasurface antennas for MIMO-OFDM

receivers with bit-limited ADCs,” IEEE Trans. Commun., vol. 69, no. 4,

pp. 2643–2659, 2021.

[29] D. R. Smith, O. Yurduseven, L. Mancera, P. Bowen, and N. Kundtz,

“Analysis of a waveguide-fed metasurface antenna,” Applied Physics,

2017.

[30] Y. Han, S. Jin, C.-K. Wen, and X. Ma, “Channel estimation for extremely

large-scale massive MIMO systems,” IEEE Wireless Commun. Lett.,

vol. 9, no. 5, pp. 633–637, 2020.

[31] W. Huang, C. Li, Y. Zeng, C. Kai, and S. He, “Near-ﬁeld full di-

mensional beam codebook design for XL-MIMO communications,” in

GLOBECOM 2023 - 2023 IEEE Global Communications Conference,

2023, pp. 4902–4908.

[32] G. Charan, M. Alrabeiah, and A. Alkhateeb, “Vision-aided 6G wireless

communications: Blockage prediction and proactive handoff,” IEEE

Trans. Veh. Tech., vol. 70, no. 10, pp. 10193–10 208, 2021.

[33] C. Julio and L. Liara, “Diagonalization and spectral decomposition of

factor block circulant matrices,” Linear Algebra and its Applications,

vol. 99, pp. 41–61, 1988.

[34] O. E. Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heath,

“Spatially sparse precoding in millimeter wave MIMO systems,” IEEE

Trans. Wireless Commun., vol. 13, no. 3, pp. 1499–1513, 2014.

[35] F. Sohrabi and W. Yu, “Hybrid digital and analog beamforming design

for large-scale antenna arrays,” IEEE Journal of Selected Topics in

Signal Processing, vol. 10, no. 3, pp. 501–513, 2016.

[36] Z. Pi, “Optimal transmitter beamforming with per-antenna power con-

straints,” in 2012 IEEE International Conference on Communications

(ICC), 2012, pp. 3779–3784.

[37] H. Wang, N. Shlezinger, Y. C. Eldar, S. Jin, M. F. Imani,

I. Yoo, and D. R. Smith, “Dynamic metasurface antennas for mimo-

ofdm receivers with bit-limited ADCs,” 2019. [Online]. Available:

https://doi.org/10.48550/arXiv.1912.06917

[38] D. W. Marquardt, “An algorithm for least-squares estimation of non-

linear parameters,” Journal of the Society for Industrial and Applied

Mathematics, vol. 11, no. 2, pp. 431–441, 1963.

[39] S. Buzzi, C. D’Andrea, T. Foggi, A. Ugolini, and G. Colavolpe, “Single-

carrier modulation versus OFDM for millimeter-wave wireless MIMO,”

IEEE Trans. Commun., vol. 66, no. 3, pp. 1335–1348, 2018.

Wei Huang (S’14-M’16) received the B.S. and M.S

degrees from the Anhui University of Science and

Technology, Huainan, China, in 2010, and 2013,

respectively, and the Ph.D. degree in information and

communication engineering with the School of In

formation Science and Engineering, Southeast Uni

versity, Nanjing, China, in 2018. Since December

2018, he has been with the School of Computer Sci

ence and Information Engineering, Hefei University

of Technology, where he is currently an Associate

Professor. His research interests include the wide

range of massive MIMO, millimeter wave MIMO wireless communication,

underwater acoustic communication, blind signal processing, sparse signal

processing, and optimization theory. He is currently involved on extremely

large-scale MIMO (XL-MIMO), vision image-aided wireless communications,

AI-based wireless communication, and dynamic metasurface antenna array.

He has served as the Guest Editor for Sensors and the Reviewer for various

journals, including the IEEE Transactions on Wireless Communications, the

IEEE Transactions on Vehicular Technology, the IEEE Wireless Communi-

cations Letters and the European Journal on Wireless Communications and

Networking. He also has been a TPC member of various conference, including

Globecom, ICC, and ICASSP, etc.

Lizheng Xu (Student Member, IEEE) received the

B.S. degree in communication engineering from An-

hui University of Technology, Maanshan, China, in

2020, and the M.S. degree in electronic information

from the Hefei University of Technology, Hefei,

China, in 2024. His research interests include near-

ﬁeld communication, structured OFDM and dynamic

metasurface antenna array.

This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and

content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3509914

Authorized licensed use limited to: HEFEI UNIVERSITY OF TECHNOLOGY. Downloaded on December 14,2024 at 09:19:53 UTC from IEEE Xplore. Restrictions apply.

Haiyang Zhang (Member, IEEE) received the B.S.

degree in communication engineering from Lanzhou

Jiaotong University, Lanzhou, China, in 2009, the

M.S. degree in information and communication en-

gineering from the Nanjing University of Posts and

Telecommunications, Nanjing, China, in 2012, and

the Ph.D. degree in information and communica-

tion engineering from Southeast University, Nanjing,

China, in 2017. He is currently a Professor with the

School of Communications and Information Engi-

neering, Nanjing University of Posts and Telecom-

munications. From 2017 to 2020, he was a Postdoctoral Research Fellow with

the Singapore University of Technology and Design, Singapore. From 2020

to 2022, he was a Postdoctoral Research Fellow with the Weizmann Institute

of Science, Israel, where he was awarded the FGS Prize for outstanding

research achievements. His research interests include 6G near-ﬁeld MIMO

communications, deep learning and sampling theory.

Chunguo Li (SM’16) received the B.S. degree in

wireless communications from Shandong University

in 2005, and Ph.D. degree in wireless communi-

cations from Southeast University in 2010. In July

2010, he joined the Faculty of Southeast University,

Nanjing China, where he was Associate Professor

between 2012 and 2016, and Full Professor since

2017 to present. From June 2012 to June 2013, he

was the Postdoctor with Concordia University, Mon-

treal, Canada. From July 2013 to August 2014, he

was with the DSL laboratory of Stanford University

as Visiting Associate Professor. From August 2017 to July 2019, he was

the adjunct professor of Xizang Minzu University under the supporting Tibet

program organized by China National Human Resources Ministry.

He is the Fellow of IET, Fellow of China Institute of Communications

(CIC), Chair of IEEE Computational Intelligence Society Nanjing Chap-

ter, and Chair of Advisory Committee for Instruments industry in Jiangsu

province. He has served as editor for a couple of international journals and

as session chair for many international conferences. His research interests

are in 6G cell-free distributed MIMO wireless communications, information

theories, and AI based audio signal processing.

Caihong Kai (Member, IEEE) received the B.S.

degree from the Hefei University of Technology,

Hefei, China, in 2003, the M.S. degree in elec

tronic engineering and computer science from the

University of Science and Technology of China,

Hefei, in 2006, and the Ph.D. degree in information

engineering from The Chinese University of Hong

Kong, Hong Kong, China, in 2010. She is currently

a Professor with the School of Computer Science

and Information Engineering, Hefei University of

Technology. Her research interests include wireless

communication and networking, network protocols, AI-aided wireless com-

munication and performance evaluation.

Yongming Huang (Senior Member, IEEE) received

the B.S. and M.S. degrees from Nanjing University,

Nanjing, China, in 2000 and 2003, respectively,

and the Ph.D. degree in electrical engineering from

Southeast University, Nanjing, in 2007.

Since March 2007, he has been on the faculty

of the School of Information Science and Engineer-

ing, Southeast University, where he is currently a

full professor. From 2008 to 2009, he was visiting

the Signal Processing Laboratory, Royal Institute

of Technology (KTH), Stockholm, Sweden. He has

been the Director of the Pervasive Communication Research Center, Purple

Mountain Laboratories, since 2019. He has published over 200 peer-reviewed

articles and holds over 80 invention patents. His current research interests

include intelligent 5G/6G mobile communications and millimeter wave wire-

less communications. He submitted around 20 technical contributions to IEEE

standards, and was awarded a certiﬁcate of appreciation for outstanding

contribution to the development of IEEE standard 802.11aj. He served as an

Associate Editor for IEEE Transactions on Signal Processing and the Guest

Editor for IEEE Journal on Selected Areas in Communications. He is currently

the Editor-at-Large of the IEEE Open Journal of the Communications Society.

This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and

content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3509914

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Article

Aug 2023

This paper provides a deterministic channel model for a scenario where wireless connectivity is established through a reflection off a smooth planar surface of an infinite extent. The developed model is rigorously built upon the physics of wave propagation and is as precise as tight are the unboundedness and smoothness assumptions on the surface. This model allows establishing how line-of-sight multiantenna communication is altered by a reflection off an electrically large surface, a situation of high interest for mmWave and terahertz frequencies.

A Digital Frequency-Dependent I/Q Imbalance and Group Delay Estimation and Compensation Modules for mmWave Single Carrier Baseband Transceivers

Article

Jan 2022

This paper proposes the estimation and compensation modules of digital frequency-dependent (FD) I/Q imbalance and group delay for single carrier (SC) mode with 15 Gbps throughput rate in the millimeter-wave (mmWave) band. The proposed compensation module of a digital recursive filter can mitigate FD I/Q imbalance effects caused by the -3 dB corner imbalance of the reconstruction and anti-aliasing filters of I/Q branches. The proposed recursive filter with the appropriate initial coefficients based on the result of Golay sequence estimation can shorten the convergence time and precisely acquire the desired coefficients. The average image rejection ratio (IRR) performance after compensating by the proposed module can be improved from 21.97 dB to 47.54 dB. Besides, the group delay distortion in reconstruction and anti-aliasing filters will cause intersymbol interference to distort the in-band signal. We employ a digital equalization filter with the filter coefficients derived from the correlation results based on the Golay sequence to compensate the in-band group delay distortion. After the proposed group delay is pre-compensation at TX, the system-level performance of error vector magnitude (EVM) performance is improved by 6.21 dB to -27.78 dB to satisfy the IEEE 802.11ad/ay requirement in 64-QAM mode. The inner RX BER performance with the FD I/Q imbalance and the group delay compensation modules can achieve 3 × 10 ^-3 at 23 dB and also satisfy the system requirement at SNR of required 25.52 dB. A four-times parallelism architecture is proposed to work at a clock rate of 625 MHz with 64-QAM mode. The gate count and power consumption of the proposed design are 154.4 K and 16.2 mW, respectively.

Near-Field Channel Estimation for Extremely Large-scale MIMO with Hybrid Precoding

Conference Paper

Dec 2021

Structured OFDM Modulation for XL-MIMO System with Dual-Wideband Effects

Abstract and Figures

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