# Direct observation of the exchange anisotropy in the helimagnetic insulator
Cu2OSeO3

Priya R. Baral<EMAIL_ADDRESS><EMAIL_ADDRESS>Crystal
Growth Facility, Institute of Physics, École Polytechnique Fédérale de
Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Chair of Computational
Condensed Matter Physics, Institute of Physics, École Polytechnique Fédérale
de Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Laboratory for Neutron
Scattering and Imaging (LNS), Paul Scherrer Institute (PSI), CH-5232 Villigen,
Switzerland  Oleg I. Utesov  Center for Theoretical Physics of Complex
Systems, Institute for Basic Science, Daejeon 34126, Republic of Korea  Chen
Luo  Helmholtz-Zentrum Berlin für Materialien und Energie, D-14109 Berlin,
Germany  Florin Radu  Helmholtz-Zentrum Berlin für Materialien und Energie,
D-14109 Berlin, Germany  Arnaud Magrez  Crystal Growth Facility, Institute of
Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland  Jonathan S. White  Laboratory for Neutron Scattering and Imaging
(LNS), Paul Scherrer Institute (PSI), CH-5232 Villigen, Switzerland  Victor
Ukleev  Helmholtz-Zentrum Berlin für Materialien und Energie, D-14109 Berlin,
Germany  Swiss Light Source, Paul Scherrer Institute (PSI), CH-5232 Villigen,
Switzerland

###### Abstract

The helical magnetic structures of cubic chiral systems are well-explained by
the competition among Heisenberg exchange, Dzyaloshinskii-Moriya interaction,
cubic anisotropy, and anisotropic exchange interaction (AEI). Recently, the
role of the latter has been argued theoretically to be crucial for the low-
temperature phase diagram of the cubic chiral magnet Cu2OSeO3, which features
tilted conical and disordered skyrmion states for a specific orientation of
the applied magnetic field ($\mu_{0}\vec{\mathrm{H}}\parallel[001]$). In this
study, we exploit transmission resonant x-ray scattering ($t-$REXS) in vector
magnetic fields to directly quantify the strength of the AEI in Cu2OSeO3, and
measure its temperature dependence. We find that the AEI continuously
increases below 50 K, resulting in a conical spiral pitch variation of $10\%$
in the (001) plane. Our results contribute to establishing the interaction
space that supports tilted cone and low-temperature skyrmion state formation,
facilitating the goals for both a quantitative description and eventual design
of the diverse spiral states existing amongst chiral magnets.

exchange anisotropy, helimagnetism, chiral magnets, skyrmions

In recent years, skyrmions in magnetic materials have attracted significant
interest due to their potential spintronic functionalities that promise a
paradigm shift in magnetic random access memory, data storage technologies,
energy saving, and unconventional computing [1, 2, 3]. Skyrmions are typically
found in thin films with asymmetric interfaces [4] and bulk noncentrosymmetric
crystals, such as chiral and polar helimagnets [5, 6].

The ground-state helical magnetic structures of cubic chiral systems are well-
described by the Bak-Jensen model, which considers the interplay between
Heisenberg exchange interaction, Dzyaloshinskii-Moriya interaction (DMI),
anisotropic exchange interaction (AEI), and cubic anisotropy (CA) [7, 8, 9].
The orientation of the helix axis is determined by a subtle interplay among
DMI, AEI, and CA. The AEI has been broadly neglected due to its weak impact on
experimental observations. However, both cubic and exchange anisotropies play
a crucial role in determining the propagation direction of the helix [9], and
ultimately, the orientation of any field-induced skyrmion lattice (SkL) in
these materials [10, 11, 12]. Moreover, in centrosymmetric materials the
competition between AEI and single-ion anisotropy can stabilize SkL even
without DMI [13].

Figure 1: (a) Illustration of the spiral modulation vector $Q$ dependence on
azimuthal angle in (110) plane for positive and negative signs of the exchange
anisotropy constant, $F_{\textrm{AEI}}$. (b) Sketch of the geometry of the
$t-$REXS experiment at VEKMAG [14]. The magnetic field was vectorially varied
in $x-y$ plane. (c) $t-$REXS patterns measured in conical states for different
azimuthal angles, $\psi=3^{\circ},72^{\circ},123^{\circ}$ at $T=14$ K. Sum of
the $t-$REXS patterns over all measured $\psi$ angles from 0 to 180∘ at (d) 14
K, (e) 25 K and (f) 50 K. Figure 2: Polar plots of the extracted conical
spiral wavevector $Q$ as a function of angle $\psi$ at (a) 14 K, (b) 20 K, (c)
25 K, (d) 30 K (e) 40 K and (f) 50 K. Solid lines correspond to the fit
according to the Eq. 1 including the offset of $17^{\circ}$ between
$\psi=0^{\circ}$ and [100] axis due to imperfect sample mounting (see
Supplementary information for more details on the sample orientation [15]).
The radial scale for $Q$ is given in the panel (f) and is the same for all
panels (a–f).

In cubic chiral magnets, the nontrivial temperature evolution of anisotropic
interactions has been demonstrated in $B20$s [16, 17], $\beta$-Mn alloys [18,
19] and Zn-doped Cu2OSeO3[20]. Often, the unambiguous experimental distinction
between the effects of cubic and exchange anisotropies is challenging since
they both affect macroscopic parameters, such as the transition fields between
helical and conical states, and conical and field-polarized states [9, 21].
Even neutron scattering techniques that are sensitive to microscopic
material’s parameters, are often unable to discriminate these two interactions
without an additional theoretical model. According to phenomenological models,
a fixed sign of the cubic anisotropy constant $K_{c}>0$, ground-state helical
spirals in cubic chiral magnets propagate along $[100]$-axes in the case of a
positive AEI constant $F_{\textrm{AEI}}>0$ (e.g. in Fe0.85Co0.15Si [22]) and
along $[111]$-axes if $F_{\textrm{AEI}}<0$ (e.g. in MnSi [17]).

Notable examples in earlier work where the role of the AEI shows up clearly
include in FeGe, where the reorientation of the spiral propagation vector from
$[100]$ to $[111]$ due to a sign change of the AEI [16], and in Zn-doped
Cu2OSeO3 where a sign change of the AEI is also argued, albeit with no
reorientation of the helix due to the predominance of CA [20].

Here, we focus on pristine Cu2OSeO3, a magnetoelectric chiral magnet with
$T_{\textrm{C}}$=58 K [23, 24] which, in addition to conventional helical,
conical, and SkL phases, also features several exotic metastable states, such
as square and elongated SkL phases [25, 26]. Recently, the competition between
the cubic and exchange anisotropies was argued to be crucial for the
manifestation of unusual yet thermodynamically stable magnetic phases in
Cu2OSeO3: tilted conical spiral and disordered skyrmions that emerge at low
temperatures when a magnetic field is applied along one of the cubic axes [27,
28, 29]. Due to the magnetoelectric coupling of Cu2OSeO3 [24], its versatile
magnetic phase diagram, and the ability to train the low-temperature skyrmion
phase [26] the material is particularly interesting for exploring chiral
magnet based applications paradigms [30]. Furthermore, developing the
understanding of the fundamental mechanism of skyrmion stabilization through
anisotropy engineering paves the way for magnetic phase manipulation amongst
the known skyrmion hosts, and which can be particularly relevant for room-
temperature topological magnetic textures among noncentrosymmetric materials
with high magnetic ordering temperatures such as chiral $\beta$-Mn-type alloys
($T_{\textrm{C}}$ up to 400 K) [19] and LiFe5O8 ($T_{\textrm{C}}$$\sim$900 K)
[31]. Therefore, the unambiguous microscopic, quantitative determination of
anisotropic interactions such as the AEI in model chiral magnets such as
pristine Cu2OSeO3is highly desirable.

Here we exploit the high momentum-space resolution of transmission resonant
elastic x-ray scattering ($t$-REXS) in vector magnetic fields [32] to quantify
directly the AEI in Cu2OSeO3. We obtain the following key results. First, the
angular variation of the conical spiral pitch in the (100) plane observed by
$t$-REXS agrees with a theory allowing the quantitative extraction of the AEI.
Second, in contrast to both FeGe [16] and lightly Zn-doped Cu2OSeO3 [20], the
sign of the AEI always remains negative across the entire temperature range
below $T_{c}$. Third, the magnitude of the AEI increases continuously below 50
K, correlating with the stability window of the tilted cone and disordered
skyrmion phases. Taken together, our results implicate the thermal evolution
of the AEI and its competition with CA as determining the structure of the
phase diagram, and contribute quantitatively towards the foundation of the
theoretical modeling and manipulation of spin textures in Cu2OSeO3 and other
anisotropic chiral magnets.

In the isotropic case, the spiral propagation vector is proportional to the
ratio of the DMI and exchange, $Q_{0}\sim D/J$. When the anisotropic
interactions come into play, the conical structure becomes distorted and, in
general, contains infinite number of harmonics, and the exact solution for the
spin structure can hardly be found. However, if the characteristic helical
energy is much larger than the anisotropic contributions one can use a
perturbative approach. In our case, in order to obtain corrections to the
spiral vector, we obtain approximate solution for the sine-Gordon equation
describing in-plane magnetization component with the anisotropy-induced terms.
The latter are due to AEI, CA and easy-plane anisotropy originating from the
tensile strain of the lamella. Importantly, the leading order approximation
allows us not to take into account small local variations of the conical
angle. Details of the derivation of the following equation are given in the
Supplementary Information [15].

At high temperatures the cubic anisotropy is small [33] (it is of the fourth
order in the magnetization modulus) and we consider only the effect of AEI and
easy-plane. The result for the spiral vector reads

$\displaystyle Q$ $\displaystyle=$ $\displaystyle
Q_{0}\Biggl{\\{}1-\dfrac{F_{\textrm{AEI}}\sin^{2}{2\psi}}{4J}\Biggr{\\}}-\dfrac{JZ^{2}\cot^{2}\alpha}{2D^{3}}\sin^{2}2(\psi-\phi)$
(1) $\displaystyle-\dfrac{JZ^{2}}{8D^{3}}\sin^{4}(\psi-\phi).$

Here $\psi$ is the azimuth angle of the conical helicoid propagation vector in
the $(001)$ plane, $\alpha$ is the conical angle ($\alpha=0$ in the fully
polarized phase), $Z$ is the easy-plane anisotropy constant and $\phi$
indicates the corresponding axis direction. At small temperatures the AEI-
induced correction in the first term of Eq. (1) dominates; other terms,
including the one stemming from CA (see [15]), being less prominent. In
addition, we have tried to fit the experimental data considering the higher-
order exchange anisotropy term, and found that the result is the same within
the error bar. Therefore, it is excluded from the analysis.

A polar plot of the spiral wavevector $Q$ depending on its orientation in the
$(001)$ is shown in Fig. 1a for positive and negative $F_{\textrm{AEI}}$
constants as an example. For $F_{\textrm{AEI}}>0$ the propagation vector of
the spiral in the ground state is favored by AEI along $\langle 001\rangle$,
while for negative $F_{\textrm{AEI}}$ spirals propagate along diagonals of the
cubic lattice $\langle 111\rangle$. Experimentally, the conical propagation
vector can be determined at will azimuthally in the (001) crystal plane by a
finite vector magnetic field (Fig. 1b). The detailed description of the
samples and the measurement setup is given in the Supplementary Information
[15]. Each conical state was prepared using the following procedure. First,
the helical state was achieved by cooling the sample down to the target
temperature at zero field ($T=14$ K), followed by ramping up the magnetic
field to $70$ mT to force the sample into the field-polarized state. The field
was then ramped down to $30$ mT in order to remain inside the conical phase
for a particular $\psi$. After each acquisition, the magnetic field was again
ramped up to $70$ mT, followed by changing its in-plane direction. This
protocol was repeated for each $\psi$ ranging between 0 and 180∘ with a 3∘
step.

At each particular sample temperature, the intensity corresponding to each of
the Friedel pair of conical peaks for the measured $\psi$ was extracted and
summed up. The resulting patterns measured at the lowest (14 K), intermediate
(25 K) and highest (50 K) temperatures are shown in Figs. 1d–f. At $T=50$ K,
the intensity profile appears almost circular, but with a slight elliptical
distortion (1f). The observed ellipticity in the intensity profile is an
indication of the uniaxial anisotropy induced by strain arising from the
contacts made on the lamella sample during FIB milling [34, 35, 36]. As the
temperature decreases, the profile starts to develop subtle features along the
marked crystallographic axes. On cooling, the azimuthal scattering intensity
distribution profile deviates from the ellipticity seen at 50 K and develops
extra humps along the in-plane $[110]$ directions. This is most strongly
pronounced at the base temperature of 14 K, as shown in (1d). Concomitantly,
$|Q|$ along the $[100]$ directions is found to be the minimum, and $Q||[110]$
the maximum. Interestingly, in contrast to FeGe [32], the helical spiral was
observed to always revert the orientation of propagation back to the $[100]$
direction upon leaving the in-plane conical phase through reduction of the
field. This is another manifestation of the strong CA in Cu2OSeO3.

Figure 3: Temperature dependence of the exchange anisotropy constant
$F_{\textrm{AEI}}$ extracted from the fit of $Q(\psi)$ according to Eq. 1. The
black dashed line is a guide to the eye.

In the next step, both the radial and azimuthal profiles of the diffracted
intensity at each $\psi$ were examined. In order to only contain a single
Bragg peak, a sector box of 3∘ angular width was chosen around each. Also,
both peaks from the Friedel pair were analyzed separately, using mirror
sectors, providing us with information on $|Q|$ in all four quadrants
simultaneously. Polar plots of the extracted peak position
$Q_{\textrm{c}}(\psi)$ in Figs. 2a–f directly show the anisotropic nature of
the conical spirals in Cu2OSeO3, and how this develops on cooling. The direct
influence of temperature dependence of the AEI on $|Q_{\textrm{c}}|$ can be
seen clearly in Fig. 2. At the lowest $T=14$ K (Fig. 2a), $Q_{\textrm{c}}$
varies between 0.086 nm-1 along [100] to 0.092 nm-1 along [110] in a
monotonous fashion, as it expected according to Eq. 1. This shows that in
Cu2OSeO3 the AEI is most pronounced at low temperatures, resulting in the
conical spiral pitch variation up to 10% between the conical spirals oriented
along [100] and [110].

In order to quantify the AEI constant, $Q(\psi)$ dependencies were fitted
according to Eq. (1) (solid lines in Fig. 2). The result is shown in Fig. 3,
where $|F_{\textrm{AEI}}|$ clearly tends to monotonically increase towards low
temperatures and reaches of $F_{\textrm{AEI}}=-0.163\pm 0.012$ pJm-1 at 14 K,
and practically vanishes at 50 K. The strain-induced anisotropy terms
containing $Z$ in Eq. 1 do not show significant variation as a function of
temperature (see Supplementary Information [15]).

The sign of the AEI constant $F_{\textrm{AEI}}$ in Cu2OSeO3 is negative in the
whole temperature range, in contrast to previous results on Zn-doped Cu2OSeO3
(Fig. 3) [20]. As shown before, a few percent Zn doping can modify the
microscopic properties of pristine Cu2OSeO3 significantly [37], and our data
supports this conclusion. Moreover, this suggests that one can tune the
microscopic parameter with a small doping, and hence finely tailor the helical
(skyrmion lattice) pitch and stability windows of anisotropy-driven phases.
Importantly, at low temperatures the two systems consistently demonstrate the
strong contribution of the magnetocrystalline anisotropy that pins the spiral
wavevector along $[100]$. Nonetheless, the competition between the AEI and
cubic anisotropies favoring different orientation of magnetic spirals is known
to stabilise more unusual magnetic spiral superstructures such as tilted
conical and disordered skyrmion phases [27, 29, 10]. A fine balance between
AEI and CA is required to theoretically reproduce low-temperature magnetic
phases in Cu2OSeO3 [29]. The strong enhancement of the AEI at low temperatures
is evident from our data and provides a much needed quantitative basis for the
stability of tilted conical and disordered skyrmion states proposed by the
theory. Therefore, a chemical tuning of the AEI would be a promising approach
to stabilize new phases far below $T_{\textrm{C}}$ in other known cubic chiral
magnets.

In summary, the study of the anisotropic exchange interaction (AEI) in the
cubic chiral magnet Cu2OSeO3 using transmission resonant x-ray scattering in
vector magnetic fields has revealed that the sign of the AEI energy constant
is negative in the whole temperature range below $T_{\textrm{C}}$, and
continuously increases below 50 K, to reach $F_{\textrm{AEI}}=-0.163$ pJm-1 at
our lowest temperature of 14 K. The sign of the AEI constant is negative in
the whole temperature range, pointing to a stronger contribution of cubic
anisotropy that pins the spiral propagation vector along $[001]$. The
magnitude of $F_{\textrm{AEI}}$ is of the same order as in FeGe but with an
opposite sign. Our measurements of the strong enhancement of the AEI at low
temperatures provide a quantitative basis for phenomenological theories that
describe how competing anisotropies in chiral magnets can stabilize novel
complex spiral magnetic states such as tilted conical and disordered skyrmion
phases. Additionally, we have presented a theoretical and experimental
framework for quantifying AEI in cubic chiral magnets, and distinct it from
CA, which is valuable for comparison with $ab-initio$ theories and for
understanding the role of AEI in the emergence of skyrmions and other exotic
magnetic state. Similar approach can be further developed for broader class of
anisotropic magnets with long-periodic spin modulations stabilized by other
mechanisms, such as frustrated interactions [38, 39, 40].

## Acknowledgements

Authors thank A. Leonov for fruitful discussions, E. Deckardt, M. Bednarzik
and Th. Jung for their help in preparation of the membranes at PSI, B. Bartova
for the assistance in FIB nano-fabrication at EPFL CIME, and K. Schwarzburg
for the help with the scanning electron microscopy measurement in the Corelab
Correlative Microscopy and Spectroscopy at Helmholtz-Zentrum Berlin. The
$t-$REXS experiment was carried out at the beamline PM-2 VEKMAG at BESSY II
synchrotron as a part of the proposal 212-10682 ST. P.R.B., J.S.W., A.M., V.U.
acknowledge funding from the SNSF Project Sinergia CRSII5_171003
NanoSkyrmionics. P.R.B. also acknowledges SNSF grant no. 200020_182536
(Frustration in structures and dynamics). We acknowledge financial support for
the VEKMAG project and for the PM2-VEKMAG beamline by the German Federal
Ministry for Education and Research (BMBF 05K2010, 05K2013, 05K2016, 05K2019)
and by HZB. F.R. acknowledge funding by the German Research Foundation via
Project No. SPP2137/RA 3570. V.U. thanks the SNSF National Centers of
Competence in Research in Molecular Ultrafast Science and Technology (NCCR
MUST-No. 51NF40-183615) for the financial support.

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# Switch-based Hybrid Beamforming for Wideband Multi-carrier Communications
††thanks: This work has been supported in part by Academy of Finland under
6Genesis Flagship (grant 318927) and EERA Project (grant 332362).

Mengyuan Ma, Nhan Thanh Nguyen and Markku Juntti  Centre for Wireless
Communications (CWC), Uninvesity of Oulu, P.O.Box 4500, FI-90014, Finland
Email: {mengyuan.ma, nhan.nguyen<EMAIL_ADDRESS>
###### Abstract

Switch-based hybrid beamforming (SW-HBF) architectures are promising for
realizing massive multiple-input multiple-output (MIMO) communications systems
because of their low cost and low power consumption. In this paper, we study
the performance of SW-HBF in a wideband multi-carrier MIMO communication
system considering the beam squint effect. We aim at designing the switch-
based combiner that maximizes the system spectral efficiency (SE). However,
the design problem is challenging because the analog combing matrix elements
are binary variables. To overcome this, we propose tabu search-based (TS) SW-
HBF schemes that can attain near-optimal performance with reasonable
computational complexity. Furthermore, we compare the total power consumption
and energy efficiency (EE) of the SW-HBF architecture to those of the phase-
shifter-based hybrid beamforming (PS-HBF) architecture. Numerical simulations
show that the proposed algorithms can efficiently find near-optimal solutions.
Moreover, the SW-HBF scheme can significantly mitigate the beam squint effect
and is less affected by the number of subcarriers than PS-HBF. It also
provides improved SE and EE performance compared to PS-HBF schemes.

###### Index Terms:

Switch-based hybrid beamforming, wideband communications, multi-carrier
systems, beam squint effect, spectral efficiency, energy efficiency.

##  I Introduction

Wideband communications systems, with their large utilizable spectrum, are
promising to meet the ever-lasting escalating demand for ultra-high-speed data
rates of future 6G wireless networks [1, 2]. However, the large numbers of
antennas in millimeter wave (mmWave) and Terahertz (THz) communications
systems require large numbers of excessively high power-hungry radio frequency
(RF) chains. As a result, there could be prohibitive power consumption and
cost. Therefore, hybrid beamforming (HBF) is envisioned as a critical
technique to realize mmWave and THz communications. It can considerably reduce
the number of costly radio frequency chains and maintain the spatial
multiplexing gain [3].

In HBF, the analog beamformer can be implemented by either soft antenna
selection with variable phase-shifters or hard antenna selection using switch
networks [4] (see Fig. 1). Nevertheless, the practical realization of the
phase-shifters for high frequencies is not a trivial task [5]. Moreover, the
large number of phase-shifters may require high power consumption, degrading
the system energy efficiency (EE). Furthermore, the beam squint effect cannot
be neglected in systems employing large bandwidth and large-sized antenna
arrays, especially in phase-shift-based HBF (PS-HBF) transceivers [6, 7]. It
can significantly degrade the spectral efficiency (SE) in wideband multi-
carrier systems. To mitigate the beam squint effect, the true-time-delay (TTD)
structure can be embedded into the RF front-end [8], which, however,
inevitably causes increased power consumption to the system. In contrast, the
frequency-independent switch-based HBF (SW-HBF) is capable of alleviating the
beam squint effect without any increase in power consumption. Compared to
phase-shifters, switch networks are simple to implement, low-power, and quick
to adapt to the channel variations [5]. Nonetheless, most of the studies on
SW-HBF focus on narrowband channel models [9, 10, 11, 5]. To the best of the
authors’ knowledge, SW-HBF in frequency-selective wideband multi-carrier
systems has not been thoroughly considered in the literature.

(a) Phase shifter-based hybrid beamforming.

(b) Switch-based hybrid beamforming

Figure 1: Illustration of PS-based and SW-based hybrid combining structures.

In this paper, we investigate the potentials of SW-HBF in overcoming the beam
squint effect of a wideband multiple-input multiple-output orthogonal
frequency-division multiplexing (MIMO-OFDM) system. Specifically, we aim at
designing the SW-based combiner that maximizes the system SE. The design
problem is challenging due to the binary variable constraints and the rank
constraint of the analog beamformer. To tackle the challenges, we first
propose a tabu search (TS) algorithm that is demonstrated to find near-optimal
solutions with reasonable complexity. Then, to further refine the solution, we
employ a projected gradient ascending (PGA) algorithm to obtain a better
initial point for the TS algorithm. Furthermore, we introduce power
consumption models for the PS-HBF and the SW-HBF architectures. Finally,
intensive simulations are provided to compare the SE and EE of the SW-HBF and
PS-HBF schemes. The results show that the former is less affected by the beam
squint effect and the number of subcarriers than the latter. Moreover, the
proposed TS-based SW-HBF schemes achieve better SE and EE performance than the
PS-HBF.

##  II System Model and Problem Formulation

###  II-A Signal Model

We consider a point-to-point MIMO-OFDM system where the transmitter is
equipped with $N_{t}$ antennas and perform fully digital beamforming. The
receiver employs either PS-HBF or SW-HBF architecture with $N_{r}$ antennas
and $N_{RF}$ RF chains, as illustrated in Figs 1(a) and 1(b). Let $K$ be the
number of subcarriers, and let ${\bm{s}}_{k}\in{\mathbb{C}}^{N_{s}\times 1}$
$(N_{s}\leq N_{RF})$ be the transmitted symbol vector at the $k$th subcarrier,
${\mathbb{E}}\left[{\bm{s}}_{k}{\bm{s}}^{H}_{k}\right]={\bm{I}}_{N_{s}},\;k=1,\cdots,K$,
where ${\bm{I}}_{N_{s}}$ denotes the $N_{s}\times N_{s}$ identity matrix. The
transmitted signal vector ${\bm{x}}_{k}\in{\mathbb{C}}^{N_{t}\times 1}$ for
each subcarrier is given as

${\bm{x}}_{k}={\bm{F}}_{k}{\bm{s}}_{k},$ (1)

where ${\bm{F}}_{k}\in{\mathbb{C}}^{N_{t}\times N_{s}}$ is the digital
precoding matrix.

At the receiver, the signal vector is first combined by the analog combiner,
represented by ${\bm{W}}_{RF}\in{\mathbb{C}}^{N_{r}\times N_{RF}}$. After
discarding the CP and performing $N_{RF}$ $K$-point fast Fourier transforms
(FFTs), the combined signal is further processed at frequency domain by low-
dimensional baseband combiner ${\bm{W}}_{BB}[k]\in{\mathbb{C}}^{N_{RF}\times
N_{s}}$ for each subcarrier. Finally, the combined signal at the $k$th
subcarrier through channel ${\bm{H}}_{k}\in{\mathbb{C}}^{N_{r}\times N_{t}}$
is given as

${\bm{y}}_{k}={\bm{W}}^{H}_{k}{\bm{H}}_{k}{\bm{F}}_{k}{\bm{s}}_{k}+{\bm{W}}^{H}_{k}{\bm{n}}_{k},$
(2)

where ${\bm{W}}_{k}={\bm{W}}_{RF}{\bm{W}}_{BB}[k]$, and
${\bm{n}}_{k}\sim\mathcal{N}(\bm{0},\sigma_{n}^{2}{\bm{I}}_{N_{r}})$ is the
additive white Gaussian noise vector at the $k$th subcarrier with
$\sigma_{n}^{2}$ being the noise variance.

###  II-B Beam Squint Effect and Channel Model

####  II-B1 Beam Squint Effect

In the conventional narrowband communications systems with analog beamforming,
the phase values of variable phase-shifters are generally optimized for the
carrier frequency. This frequency-dependent design incurs the beam squint
effect when it is applied to wideband multi-carrier systems [6]. Specifically,
there may be considerable performance loss for frequencies other than the
carrier frequency due to beam patterns of analog beamformers vary with
frequencies. Fig. 2 illustrates the beam patterns as a function of beam focus
angle $\phi$ for different frequencies. It can be observed that when the
beamformer points to angle $\phi_{0}=\pi/6$ at carrier frequency
$f_{c}=60\rm{GHz}$, the beamforming gain at other frequencies suffer a
significant loss in that their beamforming focus angles squint away from
$\pi/6$.

Figure 2: Illustration of beam squint effect in multi-carrier systems with
beam focus $\phi_{0}=\pi/6$, $N=64$ antennas, uniform linear array with
antenna sapcing $d_{s}=\lambda_{c}/2$, carrier frequency $f_{c}=60\rm{GHz}$,
bandwidth $B=4\rm{GHz}$.

####  II-B2 Wideband Channel Model with Beam Squint Effect

Since the beam squint effect is essentially induced by the frequency-dependent
beam steering vectors, we adopt the modified channel model [12], which
incorporates the frequency dependency into the classical geometric channel
model [13, 14, 15]. Assuming uniform linear array (ULA) is utilized, the
$d$-th tap of the channel at frequency $f$ can be modeled as [12]

${\bm{H}}_{f}[d]=\sum_{l=1}^{L}\alpha_{l}p\left(dT_{s}-\tau_{l}\right){\bm{a}}_{r}\left(\theta_{l}^{r},f\right){\bm{a}}_{t}^{H}\left(\theta_{l}^{t},f\right),$
(3)

where $L$ is the number of distinct scattering clusters,
$\alpha_{l}\sim\mathcal{C}\mathcal{N}(0,1)$ and $\tau_{l}$ are the complex
gain and the delay of the $l$th cluster, respectively, $\theta_{l}^{r}$ and
$\theta_{l}^{t}$ represent the angle of arrival (AoA) and angle of departure
(AoD) of the $l$th cluster, respectively, and finally $p(\tau)$ denotes the
pulse-shaping filter for $T_{s}$-spaced signaling evaluated at $\tau$ seconds
[13]. The transmit/receive steering vector at frequency $f$ is given by

${\bm{a}}(\theta,f)=\left[1,e^{-j2\pi\psi(\theta)\frac{f}{f_{c}}},\cdots,e^{-j2\pi(N-1)\psi(\theta)\frac{f}{f_{c}}}\right]^{T},$
(4)

where $N\in\\{N_{t},N_{r}\\}$ is the number of antennas,
$\psi(x)\triangleq\frac{d_{s}\sin(x)}{\lambda_{c}}$ with $\lambda_{c}$ being
the wavelength of carrier frequency, and $d_{s}$ being the antenna spacing
distance. The frequency-domain channel at the $k$th subcarrier,
$k=1,\cdots,K$, can be expressed as [13]

${\bm{H}}_{k}=\sum_{d=0}^{D-1}{\bm{H}}_{f_{k}}[d]e^{-j\frac{2\pi k}{K}d},$ (5)

where $f_{k}$ denotes the central frequency of the $k$th subcarrier with
bandwidth $B$, which can be described as [12]

$f_{k}=f_{c}+\left(k-\frac{K+1}{2}\right)\frac{B}{K},\quad\forall k.$ (6)

###  II-C Problem Formulation

Assuming the availability of full channel state information, we aim at
designing the combiner of SW-HBF that maximizes the system SE of the
considered wideband MIMO-OFDM system. For the PS-HBF (see Fig. 1(a)), the set
of feasible analog combining vectors (i.e., the columns of ${\bm{W}}_{RF}$) is
given by $\mathcal{U}_{1}=\left\\{{\bm{u}}\in{\mathbb{C}}^{N_{r}\times
1}\left||u_{j}|=1,j=1,\cdots,N_{r}\right.\right\\}$, where $u_{j}$ denotes the
$j$th element of vector ${\bm{u}}$, $|x|$ denotes the modulus of a complex
number $x$. Whereas the feasible set of SW-based analog combiner in the SW-HBF
scheme (see Fig. 1(b)) is given by
$\mathcal{U}_{2}=\left\\{{\bm{u}}\in\mathcal{B}^{N_{r}\times 1}\right\\}$,
where $\mathcal{B}=\\{0,1\\}$. Assuming that the transmit symbol at each
subcarrier follows a Gaussian distribution, the problem of designing the
combiner in the PS-HBF and SW-HBF schemes can be formulated as

$\displaystyle\underset{{\bm{W}}_{RF},\atop\left\\{{\bm{F}}_{k},{\bm{W}}_{BB}[k]\right\\}_{k=1}^{K}}{\max}$
$\displaystyle\frac{1}{K}\sum_{k=1}^{K}\log_{2}\left|{\bm{I}}_{N_{s}}+\frac{1}{\sigma^{2}_{n}}{\bm{W}}_{k}^{\dagger}{\bm{H}}_{k}{\bm{F}}_{k}{\bm{F}}^{H}_{k}{\bm{H}}^{H}_{k}{\bm{W}}_{k}\right|$
(7a) $\displaystyle\rm{s.t.}\qquad$
$\displaystyle\quad\sum_{k=1}^{K}\|{\bm{F}}_{k}\|^{2}_{F}\leq P_{b},$ (7b)
$\displaystyle\quad{\bm{W}}_{RF}(:,j)\in\mathcal{U},\,j=1,\cdots,N_{RF},$ (7c)
$\displaystyle\quad{\rm rank}({\bm{W}}_{RF})\geq N_{s},$ (7d)

where $\dagger$ denotes the Moore-Penrose pseudo inversion,
${\bm{W}}_{RF}(:,j)$ denotes the $j$th column of ${\bm{W}}_{RF}$ and $P_{b}$
is the transmit power budget. Note that $\mathcal{U}=\mathcal{A}_{1}$ for PS-
HBF; in contrast, for SW-HBF, $\mathcal{U}=\mathcal{A}_{2}$.

##  III SW-HBF Design

The PS-HBF design problem can be solved by the methods proposed in [16, 17].
In contrast, SW-HBF design is more challenging, and its solution is
unavailable in the literature. In (7), the objective function is non-convex
due to the binary variable constraints (7c) and the rank constraint (7d) of
the analog combiner. The optimal solution can be found by the exhaustive
search, which, however, is computationally prohibitive. To overcome this, we
develop a computationally efficient solution to the SE maximization problem by
decoupling the design of $\left\\{{\bm{F}}_{k}\right\\}_{k=1}^{K}$ and
$\left\\{{\bm{W}}_{k}\right\\}_{k=1}^{K}$. Specifically, for the design of
transmit beamformers $\left\\{{\bm{F}}_{k}\right\\}_{k=1}^{K}$, we assume that
the optimal receiver is used. Then, the receive beamformers
$\left\\{{\bm{W}}_{k}\right\\}_{k=1}^{K}$ are obtained given the transmit
beamformers $\left\\{{\bm{F}}_{k}\right\\}_{k=1}^{K}$. The solutions to these
subproblems are presented in the following subsections.

###  III-A Transmit Beamformer Design

Given the receive beamforming matrices
$\left\\{{\bm{W}}_{k}\right\\}_{k=1}^{K}$, the transmit beamforming design
problem is expressed as

$\displaystyle\max\limits_{\\{{\bm{F}}_{k}\\}_{k=1}^{K}}$
$\displaystyle\quad\frac{1}{K}\sum_{k=1}^{K}\log_{2}\left|{\bm{I}}_{N_{r}}+\frac{1}{\sigma_{n}^{2}}{\bm{H}}_{k}{\bm{F}}_{k}{\bm{F}}^{H}_{k}{\bm{H}}^{H}_{k}\right|$
(8) s.t. $\displaystyle\quad\sum_{k=1}^{K}\|{\bm{F}}_{k}\|^{2}_{F}\leq P_{b}.$

Let ${\bm{H}}_{k}={\bm{U}}_{k}{\bm{\Sigma}}_{k}{\bm{V}}_{k}^{H}$ be the
singular value decomposition (SVD) of ${\bm{H}}_{k}$, where
${\bm{U}}_{k}\in{\mathbb{C}}^{N_{r}\times
N_{s}},{\bm{\Sigma}}_{k}\in{\mathbb{C}}^{N_{s}\times
N_{s}},{\bm{V}}_{k}\in{\mathbb{C}}^{N_{t}\times N_{s}}$. The optimal solution
for ${\bm{F}}_{k}$ can be given as [18]

${\bm{F}}_{k}={\bm{V}}_{k}{\bm{\Gamma}}_{k}^{\frac{1}{2}}{\bm{B}}_{k},$ (9)

where ${\bm{B}}_{k}$ is any $N_{s}\times N_{s}$ unitary matrix.
${\bm{\Gamma}}_{k}={\rm diag}(p_{k,1},\cdots,p_{k,N_{s}})$ (satisfying
$\sum_{k=1}^{K}Tr({\bm{\Gamma}}_{k})=P_{b}$) is the diagonal matrix obtained
by the water-filling approach, i.e.,

$p_{k,i}=\left[\mu-\frac{\sigma_{n}^{2}}{\lambda_{{\bm{H}}_{k},i}}\right]^{+},$
(10)

where $\lambda_{{\bm{H}}_{k},i}$ denotes the $i$th largest eigenvalue of
${\bm{H}}_{k}$, $\mu$ is the water level, and in (10),
$[x]^{+}\triangleq\max(x,0)$, $\forall x\in{\mathbb{R}}$.

###  III-B Receive Beamformer Design

For a fixed analog combiner ${\bm{W}}_{RF}$, the optimal digital combiner of
each subcarrier is the MMSE solution [16],

${\bm{W}}_{BB}[k]=\left({\bm{J}}_{k}{\bm{J}}^{H}_{k}+\sigma_{n}^{2}{\bm{W}}_{RF}^{H}{\bm{W}}_{RF}\right)^{-1}{\bm{J}}_{k},$
(11)

where ${\bm{J}}_{k}\triangleq{\bm{W}}_{RF}^{H}{\bm{H}}_{k}{\bm{F}}_{k}$. Since
the optimal MMSE digital combiner can achieve maximum SE , the problem of
designing analog combiner is expressed as [19]

$\displaystyle\underset{{\bm{W}}_{RF}}{\max}$
$\displaystyle\quad\frac{1}{K}\sum_{k=1}^{K}\log_{2}\left|{\bm{I}}_{N_{RF}}+\frac{1}{\sigma^{2}_{n}}{\bm{W}}_{RF}^{\dagger}\tilde{{\bm{F}}}_{k}{\bm{W}}_{RF}\right|$
(12a) $\displaystyle\rm{s.t.}$
$\displaystyle\quad{\bm{W}}_{RF}(:,j)\in\mathcal{U}_{2},\,j=1,\cdots,N_{RF},$
(12b) $\displaystyle\quad{\rm rank}({\bm{W}}_{RF})\geq N_{s},$ (12c)

where
$\tilde{{\bm{F}}}_{k}\triangleq{\bm{H}}_{k}{\bm{F}}_{k}{\bm{F}}^{H}_{k}{\bm{H}}^{H}_{k}$.
Problem (12) is still challenging due to the binary variable constraints (12b)
and rank constraint (12c). The optima can be found by exhaustive search, which
is prohibitively complex. To solve the problem efficiently, we develop a low
complexity algorithm based on tabu search (TS) method [20].

TS algorithm begins to search for the neighbors of an initial point
${\bm{w}}_{0}$ and records the best neighbor with the largest objective value
$f({\bm{w}}_{b})$ as the best candidate ${\bm{w}}_{b}$. It then iteratively
searches for the neighbors of the best candidate ${\bm{w}}_{b}$ and updates
${\bm{w}}_{b}$ until the stopping criteria are met. During the process, a tabu
list $\mathcal{L}$ of length $L$ is used to record the visited points to avoid
cycling. The output of the TS algorithm is the best solution
${\bm{w}}_{b}^{*}$ found in the iterations, which achieves the largest value
of the objective function.

For problem (12), the objective function can be defined as

$f({\bm{W}}_{RF})=\frac{1}{K}\sum_{k=1}^{K}\log_{2}\left|{\bm{I}}_{N_{RF}}+\frac{1}{\sigma^{2}_{n}}{\bm{W}}_{RF}^{\dagger}\tilde{{\bm{F}}}_{k}{\bm{W}}_{RF}\right|.$
(13)

For notation convenience, we will use ${\bm{W}}$ in the following part to
represent ${\bm{W}}_{RF}$. Let
${\bm{e}}=[e_{1},\cdots,e_{N_{r}N_{RF}}]^{T}\in\mathcal{B}^{N_{r}N_{RF}}$, and
${\bm{e}}_{i}$ be the vector in which $i$th element is $1$ and other elements
are $0$, i.e., $e_{i}=1,e_{j\neq i}=0,\forall j$. Let ${\bm{W}}_{b}$ be the
found best candidate. Furthermore, let ${\bm{w}}_{b}\triangleq{\rm
vec}({\bm{W}}_{b})$ be the vectorization of ${\bm{W}}_{b}$. The neighbor set
of ${\bm{w}}_{b}$ is given by

$\displaystyle\mathcal{N}({\bm{w}}_{b})=$
$\displaystyle\\{{\bm{w}}\in\mathcal{B}^{N_{r}N_{RF}}\left||{\bm{w}}-{\bm{w}}_{b}|={\bm{e}}_{i},{\rm
rank}({\bm{W}})\geq N_{s},\right.$ $\displaystyle i=1,\cdots,N_{r}N_{RF}\\},$
(14)

where ${\bm{W}}\triangleq{\rm vec}^{-1}({\bm{w}})$, $|{\bm{x}}|$
(${\bm{x}}\in\mathcal{B}^{N_{r}N_{RF}\times 1}$) denotes the vector which has
element-wise absolute value of ${\bm{x}}$. The TS algorithm solving problem
(12) is summarized in Algorithm 1 where ${\bm{W}}_{0}$ is the initial
candidate, and $N_{iter}$ is the maximum iteration.

Input: $L,{\bm{W}}_{0},N_{iter},i=0$

Output: ${\bm{W}}_{b}^{*}={\rm vec}^{-1}({\bm{w}}_{b}^{*})$

1

2${\bm{w}}_{0}={\rm
vec}({\bm{W}}_{0}),{\bm{w}}_{b}\leftarrow{\bm{w}}_{0},{\bm{W}}_{b}\leftarrow{\bm{W}}_{0},{\bm{w}}_{b}^{*}\leftarrow{\bm{w}}_{0}$;

3 $\mathcal{L}=\varnothing,\mathcal{L}\leftarrow\mathcal{L}\cup{\bm{w}}_{0}$ ;

4 while  _$i\leq N_{iter}\;\ &$ not converge_ do

5

6 for  _${\bm{w}}\in\mathcal{N}({\bm{w}}_{b})$ _ do

7 ${\bm{W}}={\rm vec}^{-1}({\bm{w}})$;

8 if  _${\bm{w}}\notin\mathcal{L}\,\ &\,f({\bm{W}})>f({\bm{W}}_{b})$ _ then

9 ${\bm{w}}_{b}\leftarrow{\bm{w}}$;

10 end if

11

12 end for

13 ${\bm{W}}_{b}={\rm vec}^{-1}({\bm{w}}_{b}),{\bm{W}}_{b}^{*}={\rm
vec}^{-1}({\bm{w}}_{b}^{*})$;

14 if  _$f({\bm{W}}_{b}) >f({\bm{W}}_{b}^{*})$ _ then

15 ${\bm{w}}_{b}^{*}\leftarrow{\bm{w}}_{b}$;

16 end if

17 $\mathcal{L}\leftarrow\mathcal{L}\cup{\bm{w}}_{b}$;

18

19 if  _$|\mathcal{L}| >L$ _ then

20 Remove the earliest ${\bm{w}}_{b}$ in $\mathcal{L}$;

21 end if

22 $i\leftarrow i+1$;

23

24 end while

Algorithm 1 TS algorithm for solving problem (12)

###### Remark 1

The convergence of the TS algorithm is guaranteed by iteratively moving to the
best neighbor with an equal or larger objective value from an initial point.
Therefore, it has the potential to find the near-optimal solution, which can
be corroborated by the results shown in Section V.

In Algorithm 1, line 1 initializes the best candidate as the chosen point
${\bm{W}}_{0}$, which is then added to the tabu list (line 2). Based on the
initial candidate ${\bm{W}}_{0}$, the procedure iteratively searches for the
best neighbor and updates the best candidate until convergence (lines 3-19).
In each iteration, the TS algorithm first collects the neighbor set
$\mathcal{N}({\bm{w}}_{b})$ and treats each neighbor
${\bm{w}}\in\mathcal{N}({\bm{w}}_{b})$ as the potential candidate. By
comparing the objective value of all potential candidates to that of the
previous best candidate, the new best candidate, which is not in the tabu
list, is found (see lines 4-9). Afterward, the procedure updates the best
solution ${\bm{w}}_{b}^{*}$ via comparing it with the best candidate
${\bm{w}}_{b}$ (lines 10-13). Finally, the best candidate ${\bm{w}}_{b}$ is
added to the tabu list to avoid repeated cycling of future search (lines
14-17).

Since the TS algorithm adopts the local search procedure, the initial search
point can greatly impact its performance and computational complexity. Thus,
we further develop a heuristic algorithm to improve the quality of the initial
point of the TS algorithm. By removing the rank constraint (12c) and relaxing
the binary variable constraints (12b), the problem is recast as

$\displaystyle\underset{{\bm{W}}}{\max}$ $\displaystyle\quad
f({\bm{W}})=\frac{1}{K}\sum_{k=1}^{K}\log_{2}\left|{\bm{I}}+\frac{1}{\sigma^{2}_{n}}{\bm{W}}^{\dagger}\tilde{{\bm{F}}}_{k}{\bm{W}}\right|$
(15) $\displaystyle\rm{s.t.}$ $\displaystyle\quad{\bm{W}}\in\mathcal{W},$

where $\mathcal{W}=\\{{\bm{W}}|{\bm{W}}(i,j)\in[0,1],\forall i,j\\}$,
${\bm{W}}(i,j)$ denotes the element of ${\bm{W}}$ at $i$th row and $j$th
column. Given an arbitrary initial point, problem (15) can be efficiently
solved by the projected gradient ascending (PGA) algorithm, which is
summarized in Algorithm 2. $[\cdot]_{\mathcal{W}}$ denotes the projection into
$\mathcal{W}$.

|  Algorithm 2: PGA algorithm for problem (15)
---|---
| 1\. Initialize: $i=1,{\bm{W}}^{i}\in\mathcal{W},c=1$
| 2\. Repeat.
| 3\. $\alpha=\frac{c}{\sqrt{i+1}}$.
| 4\.
$\mathbf{W}^{i+1}\leftarrow[\mathbf{W}^{i}+\alpha\nabla_{{\bm{W}}^{i}}f({\bm{W}}^{i})]_{\mathcal{W}}$.
| 5\. $i=i+1$.
| 6\. Until convergence.
| 7\. Output: ${\bm{W}}_{pga}$.

Let ${\bm{W}}_{pga}$ denote the output of the PGA algorithm. The initial
search point of the TS algorithm can be obtained by rounding ${\bm{W}}_{pga}$
to the nearest solution in the feasible space of problem (12), i.e.,

${\bm{W}}_{0}=r({\bm{W}}_{pga}),$ (16)

where $r(\cdot)$ is the rounding function. By integrating Algorithm 2 into the
Algorithm 1, we get an improved TS algorithm, which is termed as PGA-aided TS
algorithm.

##  IV Power consumption model

Based on the architectures illustrated in Fig. 1(a) and 1(b), the total power
consumption of the fully digital beamforming (DBF), PS-HBF, and SW-HBF schemes
are given as

$\displaystyle P^{\rm DBF}_{\rm total}$
$\displaystyle=N_{r}(P_{LNA}+P_{RF}+2P_{ADC}),$ (17) $\displaystyle P^{\rm PS-
HBF}_{\rm total}$ $\displaystyle=N_{r}(P_{LNA}+P_{SP}+N_{RF}P_{PS})$
$\displaystyle\qquad\qquad+N_{RF}(P_{RF}+P_{C}+2P_{ADC}),$ (18) $\displaystyle
P^{\rm SW-HBF}_{\rm total}$ $\displaystyle=N_{r}(P_{LNA}+P_{SP}+N_{RF}P_{SW})$
$\displaystyle\qquad\qquad+N_{RF}(P_{RF}+P_{C}+2P_{ADC}),$ (19)

respectively, where $P_{RF}$ represents the power consumption of an RF chain,
which can be given as

$P_{RF}=P_{M}+P_{LO}+P_{LPF}+P_{BBamp},$ (20)

where $P_{M}$, $P_{LO}$, $P_{LPF}$, and $P_{BBamp}$ are the power consumption
of the mixer, the local oscillator, the low pass filter, and the baseband
amplifier, respectively. As a result, the system EE is defined as

$EE=\frac{SE}{P_{\rm total}},$ (21)

where $EE$ and $SE$ represent the energy efficiency and spectral efficiency,
respectively.

##  V Simulation Results

In this section, we present the numerical results to evaluate the performance
and computational complexity of the proposed SW-HBF schemes in the considered
system. In the simulations, we use the channel model given in Section II-B
with $L=10,d_{s}=\frac{\lambda_{c}}{2},f_{c}=60{\rm GHz},B=1{\rm GHz},K=64$.
The AoA/AoDs are uniformly distributed over $[0,2\pi)$, and the pulse shaping
filter is modeled as [13]

$\displaystyle
p(t)=\begin{cases}\frac{\pi}{4}\operatorname{sinc}\left(\frac{1}{2\beta}\right),&\text{if~{}}t=\pm\frac{T_{\mathrm{s}}}{2\beta}\\\
\operatorname{sinc}\left(\frac{t}{T_{\mathrm{s}}}\right)\frac{\cos\left(\frac{\pi\beta
t}{T_{\mathrm{s}}}\right)}{1-\left(\frac{2\beta
t}{T_{\mathrm{s}}}\right)^{2}},&\text{otherwise},\end{cases}$ (22)

with $T_{s}$ the sampling period and the roll-off factor $\beta=1$. The path
delay is uniformly distributed in $[0,(D-1)T_{s}]$ where $D$ is the cyclic
prefix length, given by $D=K/4$ according to 802.11ad. The SNR is defined as
SNR$\triangleq\frac{P_{b}}{K\sigma_{n}^{2}}$. The assumptions on the component
power consumptions are given in the Table I [10, 21]. All reported results are
averaged over $10^{3}$ channel realizations.

TABLE I: Power consumption of each device Device | Notation | Value
---|---|---
Low Noise Amplifier (LNA) | $P_{LNA}$ | 39mW
Splitter | $P_{SP}$ | 19.5mW
Combiner | $P_{C}$ | 19.5mW
Phase shifter | $P_{PS}$ | 30mW
Switch | $P_{SW}$ | 5mW
Mixer | $P_{M}$ | 19mW
Local oscillator | $P_{LO}$ | 5mW
Low pass filter | $P_{LPF}$ | 14mW
Base-band amplifier | $P_{BBamp}$ | 5mW
ADC | $P_{ADC}$ | 240mW

###  V-A Performance Evaluation

Figs. 3 and 4 show the average SE and EE of different algorithms versus the
SNR. For comparison, we include the PS-HBF in large-scale antenna arrays (PS-
HBF-LSAA) algorithm in [17], and the PS-HBF with closed-form solutions (PS-
HBF-CS) in [16]. For SW-HBF, the performance of optimal solution obtained by
exhaustive search (ES) method and randomly generated solution by the random
strategy are presented. Moreover, the performance of optimal digital
beamforming (DBF) via the water-filling algorithm is also exhibited. It can be
observed from Figs. 3 and 4 that the optimal SW-HBF can achieve the best
performance in terms of SE and EE. Furthermore, the TS and PGA-aided TS
algorithms can obtain near-optimal solutions and perform better than the PS-
HBF-LSAA and PS-HBF-CS algorithms. The performance of the random solution is
considerably worse than the TS algorithm, which demonstrates the effectiveness
of the proposed TS algorithms. Finally, we can observe that the performance of
the PGA-aided TS algorithm is slightly better than that of the vanilla TS
algorithm. Based on the results shown in Figs. 3 and 4, we can conclude that
the SW-HBF scheme is able to provide better SE and EE performance than that of
PS-HBF schemes in wideband multi-carrier systems.

Figs. 5(a) and 5(b) show the system SE versus the system bandwidth and the
number of subcarriers, respectively. We can observe from Fig. 5(a) that as the
bandwidth increases, the system suffers from more severe beam squint effect,
rendering a significant loss in the SE. With the proposed TS algorithms, the
SW-HBF is less affected by the beam squint effect and can achieve higher SE
than PS-HBF. Moreover, it can be observed from Fig. 5(b) that with more
subcarriers, the use of the common analog beamformer induces larger loss of
SE. However, when there are more subcarriers, the channel correlation of
subcarriers gets larger, making the system SE less affected by the analog
beamformer. This explains the slight decrease of the SE with the increasing
number of subcarriers, as is shown in Fig. 5(b). In summary, the TS-based SW-
HBF schemes are less affected by the beam squint effect and the number of
subcarriers, and they can achieve higher SE than the PS-HBF-LSAA and PS-HBF-CS
schemes. Moreover, the performance of the PGA-aided TS algorithm is slightly
better than that of the TS algorithm without a PGA solution.

Figure 3: SE of considered HBF algorithms vs SNR with
$N_{t}=16,N_{r}=8,N_{s}=N_{RF}=2,K=64$ Figure 4: EE of the considered HBF
algorithms vs SNR with $N_{t}=16,N_{r}=8,N_{s}=N_{RF}=2,K=64$

###  V-B Complexity Analysis

The complexity of the ES method and proposed TS algorithms come from the
computation of the objective function (12a) and the rank of the potential
solution ${\bm{W}}$. Since the TS procedure dominates the complexity of the
PGA-aided TS algorithm, the complexities of the two considered TS algorithms
are approximately the same. The complexity of the ES method and the TS
algorithm are $\mathcal{O}(KN_{r}^{2}N_{RF}2^{N_{r}N_{RF}})$ and
$\mathcal{O}(N_{iter}KN_{r}^{3}N_{RF}^{2})$, respectively. As shown in Figs.
6(a) and 6(b), the proposed TS algorithms have much lower computational
complexity compared with the optimal ES method.

(a) SE vs bandwidth.

(b) SE vs the number of subcarriers.

Figure 5: SE versus system bandwidth and number of subcarriers with
$N_{t}=64,N_{r}=64,N_{s}=N_{RF}=4,f_{c}=60{\rm GHz}$.

(a) Comparison of computational complexity.

(b) Computational complexity of TS algorithm.

Figure 6: Complexity analysis.

##  VI Conclusion

In this paper, we study the performance of SW-HBF in a MIMO-OFDM system with
the frequency-selective wideband channel. The near-optimal solution to the
analog combiner that maximizes the system SE is obtained via the proposed two
TS algorithms. Furthermore, we present the power consumption model of the SW-
HBF and PS-HBF architectures. Numerical simulations compare the SE and EE
achieved by the SW-HBF to those of the PS-HBF schemes. They demonstrate that
the former is able to obtain better SE and EE and less affected by the beam
squint effect than the latter. The results show that employing SW-HBF can reap
more benefits than using PS-HBF in a wideband multi-carrier system.

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