# Truncated Beam Sweeping for Spatial Covariance Matrix Reconstruction in
Hybrid Massive MIMO

Yinsheng Liu, Hongtao Duan, and Xi Liao Yinsheng Liu is with State Key
Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University,
Beijing 100044, China, and Frontiers Science Center for Smart High-spped
Railway System (e-mail: ys.liu@bjtu.edu.cn).Hongtao Duan is with Beijing radio
monitoring station of State Radio Monitoring Center (SRMC), Beijing 100037,
China.Xi Liao is with Chongqing Key Laboratory of Complex Environmental
Communications, School of Communication and Information Engineering, Chongqing
University of Posts and Telecommunications, Chongqing 400065, China
(e-mail:liaoxi@cqupt.edu.cn).Corresponding author: Yinsheng Liu.

###### Abstract

Spatial covariance matrix (SCM) is essential in many applications of multi-
antenna systems such as massive multiple-input multiple-output (MIMO). For
massive MIMO operating at millimeter-wave bands, hybrid analog-digital
structure has been adopted to reduce the cost of radio frequency (RF) chains.
In this situation, signals received at the antennas are unavailable to the
digital receiver, and as a consequence, traditional sample average approach
cannot be used for SCM reconstruction in hybrid massive MIMO. To address this
issue, beam sweeping algorithm (BSA), which can reconstruct SCM effectively in
hybrid massive MIMO, has been proposed in our previous work. In this paper, a
truncated BSA is further proposed for SCM reconstruction by taking into
account the patterns of antenna elements in the array. Due to the directive
antenna pattern, sweeping results corresponding to predetermined direction-of-
angles (DOA) far from the normal direction are small and thus can be replaced
by predetermined constants. At the cost of negligible performance reduction,
SCM can be reconstructed efficiently by sweeping only the predetermined DOAs
that are close to the normal direction. In this way, BSA can be conducted much
faster than its traditional counterpart. Insightful analysis will be also
included to show the impact of truncation on the performance.

###### Index Terms:

millimeter-wave, massive MIMO, hybrid structure, spatial covariance matrix.

##  I Introduction

Spatial covariance matrix (SCM) is essential in many applications of multi-
antenna systems [1, 2], such as massive multiple-input multiple-output (MIMO).
Massive MIMO is one of the most important enabling technologies in 5G and its
beyond [3]. Due to a large number of antennas, massive MIMO is essential to
millimeter-wave bands because the large array gain can compensate for the high
path loss, and frequency resources at millimeter-wave bands can be therefore
exploited efficiently [4, 5].

To reduce the number of radio frequency (RF) chains, hybrid structure has been
adopted for massive MIMO operating at millimeter-wave bands [6, 7, 8]. In
hybrid systems, one RF chain is connected to multiple antennas, so that the
number of RF chains can be greatly reduced. However, in hybrid massive MIMO,
the received signals at the antennas are first fed to the analog phase
shifters and then combined in the analog domain before sent to the digital
receiver. Consequently, the received signals at the antennas are unavailable
to the digital receiver, and thus traditional sample average approach cannot
be used for SCM reconstruction [9]. To address this issue, we have developed a
beam sweeping algorithm (BSA) for SCM reconstruction in hybrid massive MIMO
systems [10]. In this approach, beam sweeping results corresponding to a group
of predetermined direction-of-angles (DOA) are collected and then SCM can be
reconstructed by solving a matrix equation. To reduce the complexity caused by
solving matrix equation in [10], a high-efficiency BSA is proposed in [11]
through apropriately adjusting the weights connected to each antenna. Also,
predetermined DOAs are carefully selected in [12] so that matrix inverse can
be completely avoided. In addition, BSA in [12] is also improved so that it
can be used in the presence of multiple RF chains under the framework of sub-
connected massive MIMO [8]. Moreover, the issue of high computational
complexity can be also addressed by adopting new computation platform, such as
quantum computer [13].

The weakness of BSA lies in that it needs a large number of beam sweeping
operations, and thus the algorithm delay caused by beam sweeping can be
significant. Although fully-connected hybrid structure can be used to reduce
algorithm delay [14], that approach is unavailable in the cases of single RF
chain or sub-connected hybrid structure. To address this issue, we propose a
truncated BSA in this paper. In this approach, patterns of antenna elements in
the array are exploited. Due to the weak response of the antenna pattern at
the DOAs far from the normal direction, the received power from those DOAs can
be very small. Therefore, sweeping results corresponding to predetermined DOAs
far from the normal direction can be truncated. Then, SCM can be efficiently
reconstructed by sweeping only the predetermined DOAs that are close to the
normal direction at the cost of negligible performance reduction. In this way,
BSA can be conducted much faster than its traditional counterpart in [10, 12].
Insightful discussion will be also presented to show the impact of truncation
on the reconstruction accuracy, as well as a theoretical analysis on the
asymptotical results which are not revealed in [10, 12].

The rest of this paper is organized as follows. In Section II, signal model
for hybrid massive MIMO is introduced, followed by a review of SCM
reconstruction issue. In Section III, truncated BSA is presented, and insights
will be shown in Section IV. Simulation results and conclusions are in Section
V and VI, respectively.

##  II System Model

###  II-A Signal Model

As in Fig. 1, consider a hybrid massive MIMO receiver composed of a uniform
linear array (ULA) with $M$ antennas. To simplify the symbol notation, a
single RF chain is considered in this paper even though the proposed approach
can be also used in the case of multiple RF chains, as in [12].

For a general model, patterns of antenna elements in the array should be taken
into account. Since all elements in the ULA are the same, all antenna elements
can share a common antenna pattern $g(\theta)$ where $\theta$ indicates the
DOA. Therefore, if denote $y_{m}(t)$ to be the received signal on the $m$-th
antenna with $m=0,1,\cdots,M-1$, then the received signal vector
$\boldsymbol{y}(t)=[y_{0}(t),y_{1}(t),\cdots,y_{M-1}(t)]^{\mathrm{T}}$ can be
represented as

$\displaystyle\boldsymbol{y}(t)=\sum_{l=0}^{L-1}\boldsymbol{a}(\theta_{l})g(\theta_{l})x_{l}(t)+\boldsymbol{z}(t),$
(1)

where $x_{l}(t)$’s ($l=0,1,\cdots,L-1)$ are $L$ signals impinging from far
field onto the array, $\theta_{l}$ is the DOA of $x_{l}(t)$, and
$\boldsymbol{z}(t)$ denotes the additive Gaussian noise vector with
$\mathrm{E}\\{\boldsymbol{z}(t)\boldsymbol{z}^{\mathrm{H}}(t)\\}=N_{0}\boldsymbol{I}$
with $N_{0}$ and $\boldsymbol{I}$ being the noise power and an identity
matrix, respectively. In (1), $\boldsymbol{a}(\theta_{l})$ is the $M\times 1$
steering vector with the $m$-th entry given by
$a_{m}(\theta_{l})=e^{j2\pi\cdot\frac{d}{\lambda}\cdot\sin\theta_{l}\cdot m},$
where $d=0.5\lambda$ is the antenna distance and $\lambda$ is the wave length.
If assuming $L$ signals are mutually independent with zero means and the power
of the $l$-th signal is $\mathrm{E}\\{|x_{l}(t)|^{2}\\}=\sigma_{l}^{2}$, then
the SCM,
$\boldsymbol{R}=\mathrm{E}\\{\boldsymbol{y}(t)\boldsymbol{y}^{\mathrm{H}}(t)\\}$,
can be obtained as

$\displaystyle\boldsymbol{R}=\sum_{l=0}^{L-1}\sigma_{l}^{2}|g(\theta_{l})|^{2}\boldsymbol{a}(\theta_{l})\boldsymbol{a}^{\mathrm{H}}(\theta_{l})+N_{0}\boldsymbol{I}.$
(2)

Note that the signal model in (1) can be also used even in the presence of
mutual coupling among antennas. As in [15], most antenna elements in a long
ULA can see the same neighboring environments, and therefore an average
antenna pattern, which is common to all antennas, can be employed to describe
the mutual coupling effect. However, due to mutual coupling, the average
antenna pattern $g(\theta)$ may deviate from the one designed for the single
antenna case. Active measurement can be used in this case to figure out the
practical $g(\theta)$.

###  II-B SCM Reconstruction

Denote $\boldsymbol{y}[n]=\boldsymbol{y}(nT_{s})$ to be the sample of received
signal where $T_{s}$ denotes the sampling period. If all entries of
$\boldsymbol{y}[n]$ are available to the digital receiver, then SCM in (2) can
be estimated using the sample average approach, that is [9]

$\displaystyle\widehat{\boldsymbol{R}}=\frac{1}{N}\sum_{n=0}^{N-1}\boldsymbol{y}[n]\boldsymbol{y}^{\mathrm{H}}[n],$
(3)

where $N$ denotes the number of samples. In this approach, received signals at
all antennas should be sent via RF chains to the digital receiver. In hybrid
massive MIMO, however, Fig. 1 shows that only the combination of the entries
of $\boldsymbol{y}[n]$ can be seen by the digital receiver because there is
only one RF chain. As a consequence, the sample average approach in (3) cannot
be used in hybrid massive MIMO systems.

To address this issue, we have developed basic BSA for SCM reconstruction in
hybrid massive MIMO with single RF chain [10]. Then, the basic BSA has been
improved in [12] to handle the case of multiple RF chains. A careful selection
of predetermined DOAs has also been presented in [12] so that the
computational complexity can be greatly reduced.

In spite of the success in SCM reconstruction, the weakness of BSA is obvious
because it needs a large number of beam sweeping operations and the algorithm
delay caused by beam sweeping can be significant. Although fully-connected
hybrid structure can be used to reduce algorithm delay [14], that approach is
unavailable in the cases of single RF chain or sub-connected hybrid structure.
To overcome this issue, a truncated BSA is proposed as we will show in Section
III.

Figure 1: Hybrid massive MIMO receiver with a single RF chain and a ULA.

##  III Truncated BSA

As the truncated BSA relies on the traditional BSA, we will first review BSA
in [10] and [12] in brief to provide a framework for the truncated BSA.

###  III-A Review on BSA

As in [10], define $\\{\theta^{(0)},\theta^{(1)},\cdots,\theta^{(Q-1)}\\}$ to
be a set of predetermined DOAs. The analog beamformers switch the beam
directions to the predetermined DOAs in turn. For the $q$-th beam, the
combination of the received signals can be represented by
$c_{q}(t)=\boldsymbol{a}^{\mathrm{H}}(\theta^{(q)})\boldsymbol{y}(t)$. From
Fig. 1, the signal combination is sampled before send to the receiver, and
thus the samples of the signal combination can be given by

$\displaystyle
c_{q}[n]=c_{q}(nT_{s})=\boldsymbol{a}^{\mathrm{H}}(\theta^{(q)})\boldsymbol{y}[n].$
(4)

Denote $P_{q}=\mathrm{E}(|c_{q}[n]|^{2})$ to be the statistical power of
$c_{q}[n]$. Using sample average, an estimation of $P_{q}$ can be obtained as

$\displaystyle\widehat{P}_{q}=\frac{1}{N}\sum_{n=0}^{N-1}|c_{q}[n]|^{2}=\boldsymbol{a}^{\mathrm{H}}(\theta^{(q)})\widehat{\boldsymbol{R}}\boldsymbol{a}(\theta^{(q)}).$
(5)

Using $\mathrm{vec}(\cdot)$ operator to (5), $\widehat{P}_{q}$ can be
rewritten as
$\widehat{P}_{q}=\boldsymbol{a}_{q}^{\mathrm{T}}\widehat{\boldsymbol{r}}$
where $\widehat{\boldsymbol{r}}=\mathrm{vec}(\widehat{\boldsymbol{R}})$ and
$\boldsymbol{a}_{q}=\boldsymbol{a}(\theta^{(q)})\otimes\boldsymbol{a}^{*}(\theta^{(q)})$
with $\otimes$ indicating Kronecker product. If taking all $Q$ predetermined
DOAs into account, we can derive that

$\displaystyle\boldsymbol{A}\widehat{\boldsymbol{r}}=\widehat{\boldsymbol{p}},$
(6)

where
$\widehat{\boldsymbol{p}}=(\widehat{P}_{0},\widehat{P}_{1},\cdots,\widehat{P}_{Q-1})^{\mathrm{T}}$
contains the estimated power on all predetermined beams and
$\boldsymbol{A}=(\boldsymbol{a}_{0},\boldsymbol{a}_{1},\cdots,\boldsymbol{a}_{Q-1})^{\mathrm{T}}$.
As in [10], unknown $\widehat{\boldsymbol{r}}$ can be obtained by solving (6)
so that the desired SCM can be reconstructed as
$\widehat{\boldsymbol{R}}=\mathrm{unvec}(\widehat{\boldsymbol{r}})$.

Direct solution of (6) suffers from high complexity due to matrix inverse. To
avoid this issue, a low-complexity BSA is investigated in [12]. In this
approach, (6) is converted to

$\displaystyle\boldsymbol{B}\widehat{\boldsymbol{\gamma}}=\widehat{\boldsymbol{p}},$
(7)

where
$\boldsymbol{B}=(\boldsymbol{b}_{0},\boldsymbol{b}_{1},\cdots,\boldsymbol{b}_{Q-1})^{\mathrm{T}}$
with $\boldsymbol{b}_{q}$ being a $(2M-1)\times 1$ vector given by

$\displaystyle\boldsymbol{b}_{q}=\left(\begin{array}[]{ccc}a_{M-1}(\theta^{(q)})&\cdots&0\\\
\vdots&&\vdots\\\ a_{0}(\theta^{(q)})&\cdots&0\\\
0&\cdots&a_{M-1}(\theta^{(q)})\\\ \vdots&&\vdots\\\
0&\cdots&a_{0}(\theta^{(q)})\\\
\end{array}\right)\boldsymbol{a}^{*}(\theta^{(q)}),$ (14)

and
$\widehat{\boldsymbol{\gamma}}=(\widehat{\gamma}[1-M],\cdots,\widehat{\gamma}[M-1])^{\mathrm{T}}$
corresponds to $2M-1$ non-repeated entries in $\boldsymbol{R}$ with
$\widehat{\gamma}[m-n]$ being the $(m,n)$-th entry of
$\widehat{\boldsymbol{R}}$, that is,
$\widehat{\gamma}[m-n]=[\widehat{\boldsymbol{R}}]_{m,n}$. If the predetermined
DOAs are selected as $\theta^{(q)}=2\mathrm{arcsin}(-1+2q/Q)$, then the number
of predetermined DOAs can be determined as $Q=2M-1$ and
$\boldsymbol{B}^{\mathrm{H}}\boldsymbol{B}$ becomes a diagonal matrix
$\boldsymbol{\Delta}$ with the $m$-th $(m=0,1,\cdots,2M-2)$ diagonal entry
given by

$\displaystyle[\boldsymbol{\Delta}]_{m,m}=\begin{cases}Q(m+1),&m\leq M-1\\\
Q(2M-1-m),&m>M-1\end{cases}.$ (15)

Therefore, unknown $\widehat{\boldsymbol{\gamma}}$ can be obtained as

$\displaystyle\widehat{\boldsymbol{\gamma}}=\boldsymbol{\Delta}^{-1}\boldsymbol{B}^{\mathrm{H}}\widehat{\boldsymbol{p}},$
(16)

where matrix inverse has been avoided. The desired SCM can be then
reconstructed using the estimations of the entries contained in
$\widehat{\boldsymbol{\gamma}}$.

###  III-B Truncated Beam Sweeping

In [10] and [12], omni-directional antenna elements are implicitly assumed,
and therefore full beam sweeping has to be adopted where power estimation in
(5) is conducted for each predetermined DOA. Actually, full beam sweeping can
be approximated by a truncated beam sweeping if we exploit the antenna pattern
$g(\theta)$.

To this end, we assume the number of samples is large enough. Then, the sample
average in (5) can be replaced by the statistical average and thus

$\displaystyle\widehat{P}_{q}$
$\displaystyle=P_{q}=\boldsymbol{a}^{\mathrm{H}}(\theta^{(q)})\boldsymbol{R}\boldsymbol{a}(\theta^{(q)}).$
(17)

Substitute (2) into (17), we have

$\displaystyle P_{q}$
$\displaystyle=\sum_{l=0}^{L-1}\sigma_{l}^{2}|g(\theta_{l})|^{2}|\boldsymbol{a}^{\mathrm{H}}(\theta^{(q)})\boldsymbol{a}(\theta_{l})|^{2}+MN_{0}$
$\displaystyle=\sum_{l=0}^{L-1}\sigma_{l}^{2}M^{2}|g(\theta_{l})|^{2}\left|\mathrm{sinc}\left[\pi\frac{d}{\lambda}M\left(\sin\theta_{l}-\sin\theta^{(q)}\right)\right]\right|^{2}$
$\displaystyle~{}~{}~{}~{}~{}+MN_{0}.$ (18)

When the number of antennas is large, the feature of $\mathrm{sinc}(\cdot)$
function in (III-B) indicates that only arriving signals with DOAs close to
$\theta^{(q)}$ can cause significant contribution to $P_{q}$. Therefore, if
the antenna pattern $g(\theta)$ has very weak response for DOAs that are far
from the normal direction, the summarization term in (III-B) can be
approximated by zero when $\theta^{(q)}$ is far from the normal direction. The
mathematical observation in above also coincides with the following physical
intuition: as the overall array response $g(\theta)\boldsymbol{a}(\theta)$ is
scaled by $g(\theta)$, the received power could be very weak if we point the
beam to a direction that is far from the normal direction.

Inspired by the above observation, the full beam sweeping in (5) can be
approximated by the following truncated beam sweeping operation

$\displaystyle\widehat{P}_{q}=\begin{cases}\displaystyle{\frac{1}{N}\sum_{n=0}^{N-1}|c_{q}[n]|^{2}},&q\in\mathcal{S}\\\
~{}~{}~{}~{}~{}MN_{0},&q\in\overline{\mathcal{S}}\end{cases}.$ (19)

In (19), $\mathcal{S}=\\{q|\left\lceil T/2\right\rceil\leq q<Q-\lfloor
T/2\rfloor\\}$ where $T$ is an integer ($0\leq T\leq Q-1$) indicating the
number of truncated predetermined DOAs with $\left\lceil\cdot\right\rceil$ and
$\left\lfloor\cdot\right\rfloor$ indicating round up and down to the nearest
integers, and $\overline{\mathcal{S}}$ is the complementary set of
${\mathcal{S}}$. It means in (19) that power estimation is only need for
$\theta^{(q)}$’s that are close to the normal direction, while a predetermined
constant can be used instead for $\theta^{(q)}$’s that are far from the normal
direction. Since only $Q-T$ predetermined DOAs needs power estimation,
truncated beam sweeping operation can be accelerated when $T>0$, at the cost
of negligible performance reduction as will be shown in Section IV.

Once the truncated sweeping results in (19) are available, they can be used in
(16) to reconstruct the desired SCM.

##  IV Insights

To derive insightful results in this section, we assume that
$\theta_{l}\in\\{\theta^{(0)},\theta^{(1)},\cdots,\theta^{(Q-1)}\\}$, and thus
the DOA associated with the $l$-th signal can be rewritten as
$\theta_{l}=\theta^{(q_{l})}$. In this case, the $\mathrm{sinc}(\cdot)$
function in (III-B) can be approximated by

$\displaystyle\mathrm{sinc}\left[\pi\frac{d}{\lambda}M\left(\sin\theta^{(q_{l})}-\sin\theta^{(q)}\right)\right]\approx\delta[q_{l}-q],$
(20)

where $\delta[\cdot]$ indicates Kronecker-delta function.

###  IV-A Performance Reduction

To evaluate the performance reduction caused by truncation, define
$\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\mathrm{F}}^{2}$ to be the
squared-error (SE) between desired SCM and reconstructed SCM. Then, we can
obtain

$\displaystyle\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\mathrm{F}}^{2}=\sum_{m=0}^{M-1}\sum_{n=0}^{M-1}|\widehat{\gamma}[m-n]-\gamma[m-n]|^{2},$
(21)

where $\gamma[m-n]=[\boldsymbol{R}]_{m,n}$. Recalling that $\boldsymbol{R}$
have $2M-1$ non-repeated entries in total, (21) can be rewritten as

$\displaystyle\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\mathrm{F}}^{2}$
$\displaystyle=\sum_{m=1-M}^{M-1}(M-|m|)\cdot|\widehat{\gamma}[m]-\gamma[m]|^{2}$
$\displaystyle=Q^{-1}\|\boldsymbol{\Delta}^{1/2}(\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma})\|_{2}^{2},$
(22)

where we have used the identity in (15) for the second equation. In (IV-A),
$\boldsymbol{\gamma}=({\gamma}[1-M],\cdots,{\gamma}[M-1])^{\mathrm{T}}$
contains $2M-1$ non-repeated entries in $\boldsymbol{R}$. If denote
$\boldsymbol{p}=({P}_{0},{P}_{1},\cdots,{P}_{Q-1})^{\mathrm{T}}$, then we can
derive, similar to (7) and (16), that
$\boldsymbol{B}{\boldsymbol{\gamma}}={\boldsymbol{p}}$ and
${\boldsymbol{\gamma}}=\boldsymbol{\Delta}^{-1}\boldsymbol{B}^{\mathrm{H}}{\boldsymbol{p}}$.
Using this identities and (16), (IV-A) can be rewritten as

$\displaystyle\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\mathrm{F}}^{2}=Q^{-1}(\widehat{\boldsymbol{p}}-\boldsymbol{p})^{\mathrm{H}}\boldsymbol{B}\boldsymbol{\Delta}^{-1}\boldsymbol{B}^{\mathrm{H}}(\widehat{\boldsymbol{p}}-\boldsymbol{p}).$
(23)

To simplify the expression
$\boldsymbol{B}\boldsymbol{\Delta}^{-1}\boldsymbol{B}^{\mathrm{H}}$, consider
that

$\displaystyle\boldsymbol{B}\boldsymbol{\Delta}^{-1}\boldsymbol{\Delta}=\boldsymbol{B}.$
(24)

Since $\boldsymbol{B}^{\mathrm{H}}\boldsymbol{B}=\boldsymbol{\Delta}$, (24)
can be rewritten as
$\boldsymbol{B}\boldsymbol{\Delta}^{-1}\boldsymbol{B}^{\mathrm{H}}\boldsymbol{B}=\boldsymbol{B}$
or equivalently

$\displaystyle(\boldsymbol{B}\boldsymbol{\Delta}^{-1}\boldsymbol{B}^{\mathrm{H}}-\boldsymbol{I})\boldsymbol{B}=\boldsymbol{O},$
(25)

where $\boldsymbol{O}$ is a $(2M-1)\times(2M-1)$ all zero matrix. Recalling
that
$2M-1=\mathrm{Rank}(\boldsymbol{B}^{\mathrm{H}}\boldsymbol{B})\leq\mathrm{Rank}(\boldsymbol{B})\leq
2M-1$, $\boldsymbol{B}$ is therefore a non-singular matrix and equation
$\boldsymbol{X}\boldsymbol{B}=\boldsymbol{O}$ has only one solution
$\boldsymbol{X}=\boldsymbol{O}$. Therefore, it shows in (25) that

$\displaystyle\boldsymbol{B}\boldsymbol{\Delta}^{-1}\boldsymbol{B}^{\mathrm{H}}=\boldsymbol{I}.$
(26)

As a result, we have

$\displaystyle\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\mathrm{F}}^{2}=Q^{-1}\|\widehat{\boldsymbol{p}}-\boldsymbol{p}\|_{2}^{2}.$
(27)

Then, using (III-B) and the assumption in (20), (27) can be obtained as

$\displaystyle\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\mathrm{F}}^{2}=\frac{M^{2}}{Q}\sum_{q\in\overline{\mathcal{S}}}\sum_{l=0}^{L-1}\sigma_{l}^{2}|g(\theta^{(q_{l})})|^{2}\delta[q_{l}-q].$
(28)

It shows in (28) that the $l$-th received signal can contribute to the SE only
when $q_{l}\in\overline{\mathcal{S}}$. As a result, performance reduction due
to truncated beam sweeping can be ignored because $g(\theta^{(q_{l})})$ is
very small if $q_{l}\in\overline{\mathcal{S}}$.

###  IV-B Asymptotical Analysis

To evaluate the impact of antenna patterns on the performance, asymptotical
analysis is used. In this case, the number of samples is finite, and thus
mean-SE (MSE) should be used instead. $T=0$ is considered so that no
truncation is employed. In addition to the assumption in (20), we also assume
that $x_{l}[n]=x_{l}(nT_{s})\sim\mathrm{CN}(0,\sigma_{l}^{2})$.

Under this situations, from (27), we can obtain

$\displaystyle\mathrm{E}(\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\mathrm{F}}^{2})=Q^{-1}\sum_{q=0}^{Q-1}\mathrm{E}(|\widehat{P}_{q}-P_{q}|^{2}).$
(29)

Since $\mathrm{E}(\widehat{\boldsymbol{R}})=\boldsymbol{R}$, we have

$\displaystyle\mathrm{E}(|\widehat{P}_{q}-P_{q}|^{2})$
$\displaystyle=\mathrm{E}[|\boldsymbol{a}^{\mathrm{H}}(\theta^{(q)})\widehat{\boldsymbol{R}}\boldsymbol{a}(\theta^{(q)})|^{2}]$
$\displaystyle~{}~{}~{}~{}-|\boldsymbol{a}^{\mathrm{H}}(\theta^{(q)}){\boldsymbol{R}}\boldsymbol{a}(\theta^{(q)})|^{2}.$
(30)

Then, substitute (1) into the first term in the right-side of (IV-B), and
after a long and tedious mathematical procedure, we obtain

$\displaystyle\mathrm{E}(|\widehat{P}_{q}-P_{q}|^{2})=\frac{1}{N}\left(\sum_{l=0}^{L-1}|\boldsymbol{a}^{\mathrm{H}}(\theta^{(q)})\boldsymbol{a}(\theta_{l})g(\theta_{l})|^{2}\sigma_{l}^{2}\right)^{2}+$
$\displaystyle~{}~{}\frac{2MN_{0}}{N}\sum_{l=0}^{M-1}|\boldsymbol{a}^{\mathrm{H}}(\theta^{(q)})\boldsymbol{a}(\theta_{l})g(\theta_{l})|^{2}\sigma_{l}^{2}+\frac{M^{2}N_{0}^{2}}{N},$
(31)

where we have used the identity
$\mathrm{E}(x_{1}x_{2}x_{3}x_{4})=\mathrm{E}(x_{1}x_{2})\mathrm{E}(x_{3}x_{4})+\mathrm{E}(x_{1}x_{3})\mathrm{E}(x_{2}x_{4})+\mathrm{E}(x_{1}x_{4})\mathrm{E}(x_{2}x_{3})$
for general Gaussian random variables $x_{1},x_{2},x_{3},x_{4}$[16].

Then, with the assumption in (20), (IV-B) is simplified as

$\displaystyle\mathrm{E}(|\widehat{P}_{q}-P_{q}|^{2})=\frac{M^{4}}{N}\sum_{l=0}^{L-1}\sigma_{l}^{4}|g(\theta^{(q_{l})})|^{4}\delta[q-q_{l}]+$
$\displaystyle\frac{2M^{3}N_{0}}{N}\sum_{l=0}^{L-1}\sigma_{l}^{2}|g(\theta^{(q_{l})})|^{2}\delta[q-q_{l}]+\frac{M^{2}N_{0}^{2}}{N}.$
(32)

The MSE in (29) can be therefore obtained as

$\displaystyle\mathrm{E}(\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\mathrm{F}}^{2})=\frac{M^{4}}{QN}\sum_{l=0}^{L-1}|g(\theta^{(q_{l})})|^{4}\sigma_{l}^{4}+$
$\displaystyle\frac{2M^{3}N_{0}}{QN}\sum_{l=0}^{L-1}|g(\theta^{(q_{l})})|^{2}\sigma_{l}^{2}+\frac{M^{2}N_{0}^{2}}{QN}.$
(33)

Following insights can be obtained from (IV-B): Fist, due to the antenna
pattern $g(\theta)$, arriving signals with DOAs close to the normal direction
contribute more to the MSE than the ones with DOAs far from the normal
direction. This observation can justify the truncated BSA because lost of
sweeping results at predetermined DOAs far from normal direction has little
impact on the performance. Second, the MSE can vanish as the increasing of
$N$, which coincides with the intuition and the simulation results in [10] and
[12].

##  V Simulation

Computer simulation is adopted in this section to demonstrate the truncated
BSA. Consider a ULA with $M=16$ antennas, and the distance between neighboring
antennas is $0.5\lambda$. Two signals arrives at the ULA. DOAs associated with
the first and the second signals are $0^{\mathrm{o}}$ and
$0<\theta<90^{\mathrm{o}}$, respectively. Arriving signals are assumed
independent with zero means and unit powers, and the signal-to-noise ratio is
$0$ dB. The number of predetermined DOAs is $Q=2M-1$ and the predetermined
DOAs are selected as $\theta^{(q)}=2\mathrm{arcsin}(-1+2q/Q)$. Similar to [10]
and [12], normalized SE (NSE) is used to evaluate the accuracy of
reconstructed SCM, that is
$\mathrm{NSE}=\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\mathrm{F}}^{2}\cdot\|\boldsymbol{R}\|_{\mathrm{F}}^{-2}.$
To take different antenna patterns into account, the following pattern
function is employed $g(\theta)=\mathrm{sinc}(\alpha\sin\theta),$ where
$\alpha$ is a constant that can adjust the pattern of $g(\theta)$. The number
of samples is $N=5000$, and the simulation results are averaged over $10000$
runs.

NSE versus the number of truncated predetermined DOAs are shown in Fig. 2 with
$\alpha=1$. Generally, NSE arises as the increasing of $T$ because more
predetermined beams are truncated. When the DOA associated with the second
arriving signal is small (e.g. $\theta=10^{\mathrm{o}},20^{\mathrm{o}}$),
performance reduction due to the rising of $T$ can be ignored because the
information of spatial distribution for arriving signals has been captured
even though a half of predetermined beams ($T_{\mathrm{max}}/Q\approx 0.5$)
are truncated. For larger DOAs (e.g.
$\theta=30^{\mathrm{o}},40^{\mathrm{o}},50^{\mathrm{o}}$), a significant
performance degradation can be observed if $T$ is large enough. When $T$ is
large, predetermined DOAs far from the normal direction cannot be swept and
thus a significant degradation can be observed. However, as the further
increasing of $\theta$ (e.g. $\theta=60^{\mathrm{o}},70^{\mathrm{o}}$), the
performance reduction becomes small again. This is because lost of sweeping
results at $\theta=60^{\mathrm{o}},70^{\mathrm{o}}$ causes little impact on
the performance since the antenna pattern $g(\theta)$ has very weak response
at these directions.

Figure 2: NSE versus the number of truncated predetermined DOAs with different
DOAs for the second arriving signal. In this figure, $\alpha=1$.

Fig. 3 shows the impact of different antenna patterns on the reconstruction
accuracy. The antenna patterns are adjusted by setting $\alpha=1,2,3$
respectively. The corresponding beam widths (BW) of mainlobe are
$53^{\mathrm{o}},25^{\mathrm{o}},17^{\mathrm{o}}$. When $\alpha$ is small, the
main beam of $g(\theta)$ is wide and the arriving signal at
$\theta=45^{\mathrm{o}}$ plays an important role in the SCM. As a result, when
$T$ is large enough, information on the spatial distribution cannot be
captured and thus a significant performance degradation can be observed. When
$\alpha$ is large, the main beam of $g(\theta)$ is narrow, and thus the
arriving signal at $\theta=45^{\mathrm{o}}$ contribute little to the SCM.
Therefore, lost of information on spatial distribution corresponding to
$\theta=45^{\mathrm{o}}$ causes little performance reduction.

##  VI Conclusions

In order to reduce the algorithm delay caused by full beam sweeping, we have
proposed a truncated BSA in this paper. By exploiting the feature of antenna
patterns, the predetermined DOAs that are far from the normal direction can be
replaced by a constant and therefore power estimation on corresponding beams
are not required any more. In this way, BSA approach can be accelerated at the
cost of negligible performance reduction. Insightful analysis, including the
performance degradation caused by truncation and the asymptotically analysis
on the effect of antenna pattern, have also been presented in this paper.
Simulation results have also been presented to justify the truncated BSA.

Figure 3: NSE versus the number of truncated predetermined DOAs with different
antenna patterns. In this figure, $\theta=45^{\mathrm{o}}$.

## References

  * [1] T. E. Tuncer and B. Friedlander, _Classical and Modern Direction-of-Arrival Estimation_. Academic, Orlando, 2009.
  * [2] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” _IEEE Trans. Antennas Propag._ , no. 3, pp. 276–280, Mar. 1986.
  * [3] E. G. Larsson, F. Tufvesson, O. Edfors, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” _IEEE Commun. Mag._ , vol. 52, no. 2, pp. 186–195, Feb. 2014.
  * [4] L. Liang and X. Dong, “Low-complexity hybrid precoding in massive multiuser MIMO systems,” _IEEE Wirel. Commun._ , vol. 3, no. 6, pp. 653 – 656, Dec. 2014.
  * [5] L. You, X. Q. Gao, G. Y. Li, X.-G. Xia, and N. Ma, “BDMA for millimeter-wave/Terahertz massive MIMO transmission with per-beam synchronization,” _IEEE J. Sel. Areas Commun._ , vol. 35, no. 7, pp. 1550–1563, Jul. 2017.
  * [6] 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, Mar. 2014.
  * [7] V. Venkateswaran and A. J. van der Veen, “Analog beamforming in MIMO communications with phase shift networks and online channel estimation,” _IEEE Trans. Signal Process._ , vol. 58, no. 8, pp. 4131–4143, Aug. 2010\.
  * [8] C. Lin and G. Y. Li, “Adaptive beamforming with resource allocation for distance-aware multi-user indoor Terahertz communications,” _IEEE Trans. Commun._ , vol. 63, no. 8, pp. 2985–2995, Aug 2015.
  * [9] D. G. Manolakis, _Statistical and Adaptie Signal Processing_. ARTech House, 2005.
  * [10] S. Li, Y. Liu, L. You, W. Wang, H. Duan, and X. Li, “Covariance matrix reconstruction for DOA estimation in hybrid massive MIMO systems,” _IEEE Wirel. Commun. Lett._ , vol. 9, no. 8, pp. 1196–1200, Apr. 2020.
  * [11] Y. Zhou, G. Liu, J. Li, Y. Li, S. Ye, and L. Li, “A high-efficiency beam sweeping algorithm for DOA estimation in the hybrid analog-digital structure,” _IEEE Wirel. Commun. Lett._ , vol. 10, no. 10, pp. 2323–2326, Oct. 2021.
  * [12] Y. Liu, Y. Yan, Y. Lou, W. Wang, and H. Duan, “Spatial covariance matrix reconstruction for doa estimation in hybrid massive mimo systems with multiple radio frequency chains,” _IEEE Trans. Veh. Techno._ , vol. 70, no. 11, pp. 12 185–12 190, Nov. 2021.
  * [13] F. Meng, Z. Li, X. Yu, and Z. Zhang, “Quantum algorithm for MUSIC-based DOA estimation in hybrid MIMO systems,” _Quantum Science and Technology_ , vol. 7, p. 025002, 2022.
  * [14] Z. Fu, Y. Liu, and Y. Yan, “Fast reconstruction and iterative updating of spatial covariance matrix for DOA estimation in hybrid massive MIMO,” _IEEE Access_ , vol. 8, pp. 213 206–213 214, Dec. 2020.
  * [15] W. L. Stutzman and G. A. Thiele, _Antenna Theory and Design_. John Wiley & Sons, New York.
  * [16] C. L. Nikias and A. P. Petropulu, _Higher-Order Spectra Analysis_. Englewood Cliffs, NJ, USA:Prentice-Hall, 1993\.