Channel Estimation and Hybrid Combining for Wideband Terahertz
Massive MIMO Systems
Dovelos, K., Matthaiou, M., Ngo, H-Q., & Bellalta, B. (2021). Channel Estimation and Hybrid Combining for
Wideband Terahertz Massive MIMO Systems.
IEEE Journal on Selected Areas in Communications
,
39
(6), 1604
- 1620. https://doi.org/10.1109/JSAC.2021.3071851
Published in:
IEEE Journal on Selected Areas in Communications
Document Version:
Peer reviewed version
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Download date:10. Aug. 2022
1
Channel Estimation and Hybrid Combining for
Wideband Terahertz Massive MIMO Systems
Konstantinos Dovelos, Student Member, IEEE, Michail Matthaiou, Senior Member, IEEE, Hien Quoc Ngo, Senior
Member, IEEE, and Boris Bellalta, Senior Member, IEEE
Abstract—Terahertz (THz) communication is widely consid-
ered as a key enabler for future 6G wireless systems. However,
THz links are subject to high propagation losses and inter-
symbol interference due to the frequency selectivity of the
channel. Massive multiple-input multiple-output (MIMO) along
with orthogonal frequency division multiplexing (OFDM) can
be used to deal with these problems. Nevertheless, when the
propagation delay across the base station (BS) antenna array
exceeds the symbol period, the spatial response of the BS array
varies over the OFDM subcarriers. This phenomenon, known
as beam squint, renders narrowband combining approaches
ineffective. Additionally, channel estimation becomes challenging
in the absence of combining gain during the training stage. In this
work, we address the channel estimation and hybrid combining
problems in wideband THz massive MIMO with uniform planar
arrays. Specifically, we first introduce a low-complexity beam
squint mitigation scheme based on true-time-delay. Next, we
propose a novel variant of the popular orthogonal matching
pursuit (OMP) algorithm to accurately estimate the channel
with low training overhead. Our channel estimation and hybrid
combining schemes are analyzed both theoretically and numeri-
cally. Moreover, the proposed schemes are extended to the multi-
antenna user case. Simulation results are provided showcasing the
performance gains offered by our design compared to standard
narrowband combining and OMP-based channel estimation.
Index Terms—Beam squint effect, compressive channel estima-
tion, hybrid combining, massive MIMO, planar antenna arrays,
wideband THz communication.
I. INTRODUCTION
Spectrum scarcity is the main bottleneck of current wireless
communication systems. As a result, new frequency bands
and signal processing techniques are required to deal with
the spectrum gridlock. In view of the enormous bandwidth
available at terahertz (THz) frequencies, communication over
the THz band is deemed a key technology for future 6G
wireless systems [1]. Specifically, the THz band, spanning
from 0.1 to 10 THz, offers much larger bandwidths than the
Manuscript received July 7, 2020; revised November 20, 2020, and
February 12, 2021; accepted March 1, 2021. The work of K. Dovelos and
B. Bellalta was supported by grants WINDMAL PGC2018-099959-B-I00
(MCIU/AEI/FEDER,UE), and SGR017-1188 (AGAUR). The work of M.
Matthaiou was supported by the EPSRC, U.K., under Grant EP/P000673/1
and by a research grant from the Department for the Economy Northern
Ireland under the US-Ireland R&D Partnership Programme. The work of H.
Q. Ngo was supported by the U.K. Research and Innovation Future Leaders
Fellowships under Grant MR/S017666/1.
K. Dovelos and B. Bellalta are with the Department of Information and
Communication Technologies, Pompeu Fabra University, Barcelona, Spain
(e-mail: konstantinos.dovelos@upf.edu; boris.bellalta@upf.edu).
M. Matthaiou and H. Q. Ngo are with the Institute of Electronics, Com-
munications and Information Technology (ECIT), Queen’s University Belfast,
Belfast, U.K. (e-mail: m.matthaiou@qub.ac.uk; hien.ngo@qub.ac.uk).
millimeter wave (mmWave) band. For example, the licensed
bandwidth in the mmWave band is usually up to 7 GHz whilst
that in the THz band will be at least 10 GHz [2]. According
to Friis transmission formula, though, the path loss becomes
more severe as the frequency increases, and thus THz signals
undergo higher attenuation than their mmWave and microwave
counterparts. However, thanks to the short wavelength of THz
signals, a very large number of antennas can tightly be packed
into a small area to form a massive multiple-input multiple-
output (MIMO) transceiver, hence effectively compensating
for the propagation losses by means of sharp beamforming [3].
Therefore, THz massive MIMO is expected to enable ultra-
high-speed communication systems, such as terabit wireless
personal/local area networks and femtocells [4].
Despite the promising performance gains of THz massive
MIMO systems, the wideband transmissions in conjunction
with the large array aperture might give rise to spatial-
frequency wideband (SFW) effects [5]. In this case, the channel
response becomes frequency-selective not only because of the
delay spread of the multi-path channel, but also due to the large
propagation delay across the array aperture [6]. As a result,
the response of the BS array can be frequency-dependent also
in a line-of-sight (LoS) scenario. When orthogonal frequency
division multiplexing (OFDM) modulation is employed to
combat inter-symbol interference, the spatial-wideband ef-
fect renders the direction-of-arrival (DoA) and direction-of-
departure (DoD) of the signals to vary over the subcarriers.
This phenomenon, termed beam squint, calls for frequency-
dependent beamforming/combining, which is not available in a
typical hybrid array architecture of THz massive MIMO. More
particularly, narrowband beamforming/combining approaches
can substantially reduce the array gain across the OFDM
subcarriers, hence leading to performance degradation [7].
Consequently, beam squint compensation is of paramount
importance for THz massive MIMO-OFDM systems.
Since accurate channel state information (CSI) is essen-
tial to effectively implement combining and/or beam squint
mitigation, channel estimation under SFW effects is another
important problem to address. Specifically, in the absence of
combining gain during channel estimation, the detection of the
paths present in the channel becomes challenging in the low
signal-to-noise ratio (SNR) regime. Additionally, due to the
massive number of BS antennas and the limited number of
radio frequency (RF) chains in a hybrid array architecture, the
channel estimation overhead becomes excessively large even
for single-antenna users under standard approaches, such as
the least squares (LS) method. In conclusion, THz massive
2
MIMO brings new challenges in the signal processing design,
and calls for carefully tailored solutions that take into account
the unique propagation characteristics in THz bands.
A. Prior Work
In this section, we review prior work on channel estimation
and hybrid beamforming in wideband mmWave/THz systems.
The authors in [8] proposed a novel single-carrier transmis-
sion scheme for THz massive MIMO, which utilizes minimum
mean-square error precoding and detection. Nevertheless, a
narrowband antenna aray model was considered, and hence
the SFW effect was ignored. A stream of recent papers on
wideband mmWave MIMO-OFDM systems (see [9]–[12], and
references therein) proposed methods to jointly optimize the
analog combiner and the digital precoder in order to maximize
the achievable rate under the beam squint effect. In a similar
spirit, [13] and [14] proposed a new analog beamforming code-
book with wider beams to avoid the array gain degradation due
to beam squint. These methods can enhance the achievable
rate when the beam squint effect is mild. However, their
performance becomes poor in THz MIMO systems due to the
much larger signaling bandwidth and number of BS antennas
compared to their mmWave counterparts [17]. To this end,
[15] proposed a wideband codebook for beam training for uni-
form linear arrays (ULAs) using true-time-delay (TTD) [16].
However, this design is limited to ULAs and beam alignment
without explicitely estimating the channel. From the relevant
literature on hybrid beamforming, we distinguish [17], which
proposed a TTD-based hybrid beamformer for THz massive
MIMO, however assuming ULAs and perfect CSI.
Despite the importance of channel estimation, there are only
few recent works in the literature investigating the channel
estimation problem under the spatial-wideband effect. More
particularly, the seminal paper [5] introduced the SFW for
mmWave massive MIMO systems, and proposed a channel es-
timation algorithm by capitalizing on the asymptotic properties
of SFW channels. However, the proposed algorithm relies on
multiplying the channel of an N-element uniform linear array
by an N-point discrete Fourier transform (DFT) matrix, and
hence entails high training overhead when the number of RF
chains is much smaller than the number of BS antennas. In a
similar spirit, [18] employed the orthogonal matching pursuit
(OMP) algorithm along with an energy-focusing preprocessing
step to estimate the SFW channel, while minimizing the power
leakage effect. Finally, [19] leveraged tools from compressive
sensing (CS) theory to tackle the channel estimation problem
in frequency-selective multiuser mmWave MIMO systems but
in the absence of the spatial-wideband effect.
B. Contributions
In this paper, we address the channel estimation and hybrid
combining problems in wideband THz MIMO. To this end,
we assume OFDM modulation, which is the most popular
transmission scheme over frequency-selective channels. The
main contributions of the paper are summarized as follows:
• We model the SFW effect in THz MIMO-OFDM systems
with a uniform planar array (UPA) at the BS. Note that
prior studies (e.g., [20], [21]) on mmWave/THz com-
munication with UPAs ignore the SFW effect. We next
show that frequency-flat combining leads to substantial
performance losses due to the severe beam squint effect
occuring across OFDM subcarriers, and propose a beam
squint compensation strategy using TTD [22] and virtual
array partition. The scope of the virtual array partition is
to reduce the number of TTD elements needed to effec-
tively mitigate beam squint. To this end, we derive the
wideband combiner expression for a rectangular planar
array, and establish its near-optimal performance with
respect to fully-digital combining analytically, as well as
through computer simulations.
• We propose a solution to the channel estimation problem
under the SFW effect. Specifically, by availing of the
channel sparsity in the angular domain, we first adopt a
sparse representation of the THz channel, and formulate
the channel estimation problem as a CS problem. We
then propose a solution based on the OMP algorithm,
which is one of the most common and simple greedy CS
methods. Contrary to existing works, we employ a wide-
band dictionary and show that channels across different
OFDM subcarriers share a common support. This enables
us to apply a variant of the simultaneous OMP algorithm,
coined as generalized simultaneous OMP (GSOMP),
which exploits the information of multiple subcarriers
to increase the probability of successfully recovering the
common support. We also evaluate the computational
complexity of the GSOMP to showcase its efficiency
with respect to the OMP. Numerical results show that
the propounded estimator outperforms the OMP-based
estimator in the low and moderate SNR regimes, whilst
achieving the same accuracy in the high SNR regime.
• We analyze the mean-square error of the GSOMP scheme
by providing the Cram
´
er-Rao lower bound (CRLB).
Moreover, we calculate the average achievable rate as-
suming imperfect channel gain knowledge at the BS. We
then show numerically that when the angle quantization
error involved in the sparse channel representation is
negligible, the performance of the GSOMP-based es-
timator is very close to the CRLB. Additionally, the
average achievable rate approaches that of the perfect
channel knowledge case at moderate and high SNR
values, hence corroborating the good performance of our
design. Finally, we extend our analysis to the case of a
multi-antenna user, and discuss the benefits of deploying
multiple antennas at the user side.
The rest of this paper is organized as follows: Section II in-
troduces the system and channel models. Section III describes
the hybrid combining problem under the beam squint effect,
and presents the proposed combining scheme. Section IV
formulates the channel estimation problem, introduces the
standard estimation methods, and explains the propounded al-
gorithm for estimating the SFW channel. Section V extends the
analysis to the multi-antenna user case. Section VI is devoted
to numerical simulations. Finally, Section VII summarizes the
main conclusions derived in this work.
3
BS
user
h( f )
scattering
object
(a) Uplink setup
θ
φ
x
y
(0,0)th BS
antenna
z
d
(b) Array geometry
Fig. 1: Illustration of the BS antenna array and its geometry considered in the system model.
Notation: Throughout the paper, D
N
(x) =
sin(Nx/2)
N sin(x/2)
is the
Dirichlet sinc function; A is a matrix; a is a vector; a is
a scalar; A
†
, A
H
, and A
T
are the pseudoinverse, conjugate
transpose, and transpose of A, respectively; A(i) is the ith
column of matrix A; A(I) is the submatrix containing the
columns of A given by the indices set I; |I| is the cardinality
of set I; tr{A} is the trace of A; blkdiag(A
1
, . . . , A
n
) is the
block diagonal matrix; [A]
n,m
is the (n, m)th element of ma-
trix A; F{·} denotes the continuous-time Fourier transform;
∗ denotes convolution; Re{·} is the real part of a complex
variable; 1
N×M
is the N ×M matrix with unit entries; I
N
is
the N ×N identity matrix; [v]
n
is the nth entry of vector v;
supp(v) = {n : [v]
n
6= 0} is the support of v; ⊗ denotes the
Kronecker product; is the element-wise product; kak
1
and
kak
2
are the l
1
-norm and l
2
-norm of vector a, respectively;
E{·} is expectation; and CN(µ, R) is a complex Gaussian
vector with mean µ and covariance matrix R.
TABLE I
MAIN NOTATION USED IN THE SYSTEM MODEL
Notation Description
S Number of subcarriers
f
s
Frequency of the sth subcarrier
B Total signal bandwidth
L Number of NLoS paths
α
l
(f) Frequency-selective attenuation of the lth path
τ
l
ToA of the lth path
(φ
l
, θ
l
) DoA of the lth path
τ
l,nm
Time delay to the (n, m)th BS antenna over the lth path
τ
nm
(φ
l
, θ
l
) Time delay from the (0, 0)th to the (n, m)th BS antenna
x(t) Baseband-equivalent of transmitted signal
x(f) Fourier transform of x(t)
x
l
(t) Distorted version of x(t) over the lth path
˜r
nm
(t) Passband signal received by the (n, m)th BS antenna
r
nm
(t) Baseband-equivalent of ˜r
nm
(t)
r
nm
(f) Fourier transform of r
nm
(t)
d Antenna spacing
f
c
Carrier frequency
c Speed of light
k
abs
Molecular absorption coefficient
D Distance between the BS and the user
Γ
l
(f) Reflection coefficient of the lth NLoS path
II. SYSTEM MODEL
Consider the uplink of a THz massive MIMO system where
the BS is equipped with an N ×M -element UPA, and serves a
single-antenna user as depicted in Fig 1(a); the multi-antenna
user case is investigated in Section V. The total number of BS
antennas is N
B
= NM, and the baseband frequency response
of the uplink channel is denoted by h(f) ∈ C
N
B
×1
. In the
sequel, we present the channel and hybrid transceiver models
used in this work.
A. THz Channel Model with Spatial-Wideband Effects
Due to limited scattering in THz bands, the propagation
channel is represented by a ray-based model of L + 1
rays [21], [23]. Hereafter, we assume that the 0th ray cor-
responds to the LoS path, while the remaining l = 1, . . . , L,
rays are non-line-of-sight (NLoS) paths. Specifically, each path
l = 0, . . . , L, is characterized by its frequency-selective path
attenuation α
l
(f), time-of-arrival (ToA) τ
l
, and DoA (φ
l
, θ
l
),
where φ
l
∈ [−π, π] and θ
l
∈ [−
π
2
,
π
2
] are the azimuth and
polar angles, respectively. In the far-field region
1
of the BS
antenna array, the total delay between the user and the (n, m)th
BS antenna through the lth path, τ
l,nm
, is calculated as
τ
l,nm
= τ
l
+ τ
nm
(φ
l
, θ
l
), (1)
where τ
nm
(φ
l
, θ
l
) accounts for the propagation delay across
the BS array, and is measured with respect to the (0, 0)th BS
antenna. For a UPA placed in the xy-plane (see Fig. 1(b)), we
then have [24]
τ
nm
(φ
l
, θ
l
) ,
d(n sin θ
l
cos φ
l
+ m sin θ
l
sin φ
l
)
c
, (2)
where d is the antenna separation, and c is the speed of
light. The channel frequency response is derived as follows.
Let x(t) be the baseband signal transmitted by the user, with
F{x(t)} = x(f). The passband signal, ˜r
nm
(t), received by the
(n, m)th BS antenna is written in the noiseless case as [25]
˜r
nm
(t) =
L
X
l=0
√
2Re
n
x
l
(t − τ
l,nm
)e
j2πf
c
(t−τ
l,nm
)
o
, (3)
where f
c
is the carrier frequency, x
l
(t) , x(t) ∗ χ
l
(t) is the
distorted baseband waveform due to the frequency-selective at-
tenuation of the lth path, and χ
l
(t) models the said distortion;
namely, F{χ
l
(t)} = α
l
(f) and F{x
l
(t)} = α
l
(f)x(f) [26].
1
Near-field considerations are provided in Section VI-D.
4
Next, the received passband signal ˜r
nm
(t) is down-converted
to the baseband signal r
nm
(t), which is given by
r
nm
(t) =
L
X
l=0
e
−j2πf
c
τ
l
e
−j2πf
c
τ
nm
(φ
l
,θ
l
)
x
l
(t − τ
l,nm
). (4)
Taking the continuous-time Fourier transform of (4) yields
r
nm
(f) = F{r
nm
(t)}
=
L
X
l=0
β
l
(f)e
−j2π(f
c
+f)τ
nm
(φ
l
,θ
l
)
x(f)e
−j2πf τ
l
, (5)
where β
l
(f) , α
l
(f)e
−j2πf
c
τ
l
is the complex gain of
the lth path. Lastly, collecting all r
nm
(f) into a vector
r(f) ∈ C
N
B
×1
gives the relation r(f) = h(f)x(f ), where
h(f) =
L
X
l=0
β
l
(f)a(φ
l
, θ
l
, f)e
−j2πf τ
l
(6)
is the baseband frequency response of the uplink channel, and
a(φ, θ, f) =
h
1, . . . , e
−j2π(f
c
+f)
d
c
(n sin θ cos φ+m sin θ sin φ)
,
. . . , e
−j2π(f
c
+f)
d
c
((N−1) sin θ cos φ+(M−1) sin θ sin φ)
i
T
(7)
is the array response vector of the BS. Here, the array response
is frequency-dependent due to the spatial-wideband effect.
2
We now introduce the path attenuation model. First, the so-
called molecular absorption loss is no longer negligible at THz
frequencies. Therefore, the path attenuation of the LoS path
is calculated as [27]
|β
0
(f)| = α
0
(f) =
c
4π(f
c
+ f )D
e
−
1
2
k
abs
(f
c
+f)D
, (8)
where D denotes the distance between the BS and the user, and
k
abs
(·) is the molecular absorption coefficient determined by
the composition of the propagation medium; different from
mmWave channels, the major molecular absorption in THz
bands comes from water vapor molecules [27]. For the NLoS
paths, we consider single-bounce reflected rays, since the
diffused and diffracted rays are heavily attenuated for distances
larger than a few meters [28]. To this end, the reflection
coefficient, Γ
l
(f), should be taken into account, which is
specified as [29, Eq. (2)]
Γ
l
(f) =
cos φ
i,l
− n
t
cos φ
t,l
cos φ
i,l
+ n
t
cos φ
t,l
e
−
8π
2
(f
c
+f )
2
σ
2
rough
cos
2
φ
i,l
c
2
,
(9)
where n
t
, Z
0
/Z is the refractive index, Z
0
= 377 Ω
is the free-space impedance, Z is the impedance of the
reflecting material, φ
i,l
is the incidence and reflection angle,
φ
t,l
= arcsin
n
−1
t
sin φ
i,l
is the refraction angle, and σ
rough
characterizes the roughness of the reflecting surface. The path
attenuation of the lth NLoS path is finally given by [30]
|β
l
(f)| = α
l
(f) = |Γ
l
(f)|α
0
(f), (10)
where l = 1, . . . , L.
2
If the delay across the BS array is small relative to the symbol period,
then x
l
(t−τ
l,nm
) ≈ x
l
(t−τ
l
). In this case, we have a spatially narrowband
channel with frequency-flat array response vectors, i.e., a(φ, θ, 0).
Baseband Combiner
(Digital)
Data
Streams
RF Chain
RF Chain
RF Combiner
(Analog)
Fig. 2: Illustration of the hybrid array structure considered in the system
model.
B. Hybrid Transceiver Model
Due to the frequency selectivity of the THz channel, OFDM
modulation is employed to combat inter-symbol interference.
Specifically, we consider S subcarriers over a signal band-
width B. Then, the baseband frequency of the sth subcar-
rier is specified as f
s
=
s −
S−1
2
B
S
, s = 0, . . . , S − 1.
A hybrid analog-digital architecture with N
RF
N
B
RF
chains is also considered at the BS to facilitate efficient
hardware implementation; each RF chain drives the array
through N
B
analog phase shifters, as shown in Fig. 2. The
hybrid combiner for the sth subcarrier is hence expressed as
F[s] = F
RF
F
BB
[s] ∈ C
N
B
×N
RF
, where F
RF
∈ C
N
B
×N
RF
is the
frequency-flat RF combiner with elements of constant ampli-
tude, i.e.,
1
√
N
B
, but variable phase, and F
BB
[s] ∈ C
N
RF
×N
RF
is
the baseband combiner. Finally, the post-processed baseband
signal, y[s] ∈ C
N
RF
×1
, for the sth subcarrier is written as
y[s] = F
H
[s]r[s]
= F
H
[s]
p
P
d
h[s]x[s] + n[s]
, (11)
where r[s] , r(f
s
) and h[s] , h(f
s
) are the received signal
and uplink channel, respectively, x[s] , x(f
s
) ∼ CN(0, 1) is
the data symbol transmitted at the sth subcarrier, P
d
denotes
the average power per data subcarrier assuming equal power
allocation among subcarriers, and n[s] ∼ CN(0, σ
2
I
N
B
) is
the additive noise vector.
Remark 1. A promising alternative to OFDM is single-
carrier with frequency domain equalization (SC-FDE) due
to its favorable peak-to-average power ratio (PAPR). In our
work, we exploit the inherent characteristics of THz channels,
i.e., high path loss and directional transmissions, which result
in a coherence bandwidth of hundreds of MHz [28]. Therefore,
a relatively small number of subcarriers is used, which is
expected to yield a tolerant PAPR.
III. HYBRID COMBINING
A. The Beam Squint Problem
Even for a moderate number of BS antennas, the propaga-
tion delay across the array can exceed the sampling period due
![TABLE II MAIN SIMULATION PARAMETERS [27], [28]](/figures/table-ii-main-simulation-parameters-27-28-35ohjegc.webp)
