Beam Steering for the Misalignment in UCA-Based
OAM Communication Systems
Rui Chen, Member, IEEE, Hui Xu, Marco Moretti, Member, IEEE and Jiandong Li, Senior Member, IEEE
Abstract—Radio frequency orbital angular momentum (RF
OAM) is a technique that provides extra degrees of freedom
to improve spectrum efficiency of wireless communications.
However, OAM requires perfect alignment of the transmit and
the receive antennas and this harsh precondition greatly chal-
lenges practical RF OAM applications. In this paper, we first
investigate the effect of the non-parallel misalignment on the
channel capacity of an OAM communication system equipped
with the uniform circular array (UCA) and then propose a
transmit/receive beam steering approach to circumvent the large
performance degradation in not only the non-parallel case but
also the off-axis and other general misalignment cases. The
effectiveness of the beam steering approach is validated through
both mathematical analysis and numerical simulations.
Index Terms—Orbital angular momentum (OAM), uniform
circular array (UCA), misalignment, beam steering
I. INTRODUCTION
The explosive growth of multimedia services and the advent
of new wireless paradigms such as Internet of Things (IoT)
calls for a large increase of wireless data capacity. On the
other hand, the radio frequency (RF) spectrum is limited and
the available spectral resources can not accommodate all the
requests for higher data rates. To deal with this problem,
research has explored innovative techniques and concepts
such as advanced modulation and coding schemes, cognitive
radio and multiple-input multiple-output (MIMO). Since the
discovery in 1992 that light beams with helical phase fronts
can carry orbital angular momentum (OAM) [1], a great
research effort has been focused on OAM at RF as a novel
approach for multiplexing a set of orthogonal modes on the
same frequency channel and achieve high spectral efficiencies.
Recent experiments have shown that optical OAM can
produce dramatic increases in capacity and spectral efficiency:
in [2], for example, polarization-multiplexed data streams
travelling along a beam of light with 8 different OAM modes
can reach a rate up to 2.5 Tbits (equivalent to 66 DVDs) per
second. In the same time, the famous ‘Venice experiment’
[3] has shown that OAM can be succesfully applied to RF
transmissions as first claimed in [4]. Since then, RF OAM
communications has attracted a lot of attention [5]–[12]. The
work in [5] shows that it is possible to achieve 32Gbit/s in
a wireless communication link at 28 GHz by multiplexing
8 OAM channels generated by spiral phase plates (SPPs). To
overcome the complexity of using SPP, [6], [13] and [14] have
Rui Chen, Hui Xu and Jiandong Li are with the State Key Laboratory of
Integrated Service Networks (ISN), Xidian University, Shaanxi 710071, China
(e-mail: rchen@xidian.edu.cn).
M. Moretti is with the University of Pisa, Dipartimento di Ingegneria
dell’Informazione, Italy (e-mail: marco.moretti@iet.unipi.it).
verified in theory and in practice the feasibility of employing
uniform circular array (UCA) to generate OAM waves .
One of the main obstacles for the commercial deployment of
RF-OAM systems is the severe precondition that the transmit
and receive antenna arrays need perfect alignment. If the
precondition is not accurately met, the system performance
quickly deteriorates. The authors in [7] summarize the effects
of misalignment qualitatively through simulations and exper-
iments on the SPP-based OAM system in millimeter wave
band. The performance degradation caused by non-parallel
misalignment in the UCA-based OAM system is shown in [8]
through electromagnetic simulation. Nevertheless, to the best
of our knowledge, the quantitative analysis of the misalign-
ment effects in the UCA-based OAM system has not been
performed yet.
In this paper we investigate the effect of misalignment
between the transmit and the receive UCA in a RF OAM
communication system and prove that beam steering is an
effective way to improve the system performance in case of
misalignment. In detail, we first derive the channel model of
the basic non-parallel misalignment case. Then, we analyze the
effect of oblique angle on the OAM channel capacity. At last,
we propose an OAM beam steering method to circumvent the
large performance degradation due to the non-parallel case,
the off-axis and other general cases. Mathematical analysis
and numerical results validated our proposed method.
Notation
: Upper (lower) case boldface letters are for ma-
trices (vectors); (·)
H
denotes complex conjugate transpose
(Hermitian), (·)
T
denotes transpose, |·| denotes modulus of a
complex number and ⊙ denotes Hadamard product.
II. SYSTEM MODEL
We consider a RF OAM communication system, where
the OAM beam is generated by an N-elements UCA at
the transmitter and received by another N-elements UCA at
the receiver. In practice, the perfect alignment between the
transmit UCA and the receive UCA may not be easy to realize.
Thus, the two common misalignment cases: non-parallel and
off-axis could be observed [7], [9]. Following the analysis in
Section IV, the off-axis case could be decomposed into two
non-parallel cases and, accordingly, we only delve into the
basic non-parallel case as shown in Fig.1.
A. Channel Model
In free space communications, propagation through the RF
channel leads to attenuation and phase rotation of the trans-
mitted signal. This effect is modelled by the multiplication for
2
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Fig. 1: The geometrical model of the transmit and the receive
UCA in the non-parallel misalignment case.
a complex constant h, whose value depends on the distance d
between the transmit and receive antenna [15]:
h(d) = β
λ
4πd
exp
−j
2πd
λ
, (1)
where λ is the wavelength, and λ/4πd denotes the degradation
of amplitude, and the complex exponential term is the phase
difference due to the propagation distance. The term β models
all constants relative to the antenna elements and their patterns.
Let us assume that the number of the transmit and receive
antenna elements is equal to N. Thus, according to (1)
and the geometric relationship depicted in Fig.1, the channel
coefficients from the n-th (1 ≤ n ≤ N) transmit antenna
element to the m-th ( 1 ≤ m ≤ N) receive antenna element
can be expressed as h
m,n
= h(d
m,n
), where the transmission
distance d
m,n
is calculated as
d
m,n
=
R
2
r
+ R
2
t
+ D
2
+ 2DR
r
sin θ sin α
− 2R
r
R
t
(cos φ cos θ + sin φ sin θ cos α)
1/2
, (2)
where D is the distance between the transmit and the receive
UCA centers, R
t
and R
r
are respectively the radiuses of the
transmit and the receive UCAs, φ = [2π(n − 1)/N + φ
0
]
and θ = [2π(m − 1)/N + θ
0
] are respectively the azimuthal
angles of the transmit and the receive UCAs, φ
0
and θ
0
are
the corresponding initial angles of the first reference antenna
element in both UCA, α is the oblique angle shown in Fig.1.
In the end, the channel matrix of the UCA-based free
space OAM communication system can be expressed as H =
[h(d
m,n
)]
N×N
. When α = 0, H is a circulant matrix that
can be decomposed by N-dimentional Fourier matrix F
N
as
H = F
H
N
ΛF
N
, Λ is a diagonal matrix with the eigenvalues
of H on its diagonal.
B. UCA-Based OAM Communication System
The OAM beam is generated by the UCA by feeding the
antenna elements with the same input signal, but with a
successive phase shift from element to element. Thus, after
a full turn the phase has the increment of 2πℓ, where ℓ is an
unbounded integer termed as OAM mode number [1].
Thus, generating the OAM beam with the mode number ℓ
could be formulated as
x
t
=
1
√
N
N
n=1
x(ℓ) exp
jℓ
2π(n − 1)
N
= f
H
(ℓ)x(ℓ), (3)
where x(ℓ) is the information signal to be transmitted by ℓ-
mode beam, x
t
is the generated vortex signal vector, f (ℓ) =
1
√
N
[1, T
ℓ
, ··· , T
ℓ(N−1)
], T = exp(−j
2π
N
). Accordingly, re-
ceiving the OAM beam with the mode number ℓ
′
could be
formulated as
y(ℓ
′
) =
1
√
N
N
m=1
y
r,m
exp
−jℓ
′
2π(m − 1)
N
= f (ℓ
′
)y
r
,
(4)
where y
r
= [y
r,1
, y
r,2
, ··· , y
r,N
]
T
is the received vortex
signal vector, and y(ℓ
′
) is the ℓ
′
-mode OAM despiralized
information signal. Then, the orthogonality between OAM
modes could be revealed by
y(ℓ
′
) = f (ℓ
′
)f
H
(ℓ)x(ℓ) =
x(ℓ) ℓ
′
= ℓ
0 ℓ
′
= ℓ.
(5)
Therefore, the transmission of N modes-multiplexed OAM
beams in the free space channel H drives the despiralized
information signal vector y to take the form
y = F
N
HF
H
N
x + n
, (6)
where y = [y(1), y(2), ··· , y(N)]
T
, x = [x(1), x(2), ··· ,
x(N)]
T
, F
N
= [f
H
(1), 2
H
(ℓ), ··· , f
H
(N)]
H
is Fourier ma-
trix, and n is the complex Gaussian noise vector with zero
mean and covariance matrix σ
2
n
I
N
. In the case of perfect
alignment, with the circulant matrix decomposition, (6) could
be further simplified as [15]
y = Λx +
˜
n, (7)
where
˜
n = F
N
n. The equation (7) shows that there is not any
inter-mode interference at the receiver. Thus, in contrast to MI-
MO system, OAM system doesn’t need complex equalization
with the channel information [10]. However, once the perfect
alignment is not met, the decomposition of H doesn’t hold,
which results in the inter-mode interference at the receiver
and thus poor system performance. Moreover, we find that
the performance of the single-mode OAM system degrades
severely even if α has a small value. Therefore, to isolate
the impact of inter-mode interference while simplifying the
analysis, we consider the OAM system with only one single
mode in this paper.
For the transmission of the ℓ-mode OAM beam, the despi-
ralized information signal y(ℓ) can be written as
y(ℓ) = f(ℓ)
Hf
H
(ℓ)x + n
= h
N
eff
(ℓ)x + ˜n, (8)
where h
N
eff
(ℓ) = f(ℓ)Hf
H
(ℓ) is the ℓ-mode effective OAM
channel. Since f (ℓ) doesn’t change the noise energy, ˜n is also
complex Gaussian variable with zero mean and variance σ
2
n
.
Thus, the channel capacity of the ℓ-mode OAM communica-
tion system described in (8) can be formulated as
C
N
(ℓ) = log
2
1 +
P
ℓ
|h
N
eff
(ℓ)|
2
σ
2
n
bit/sec/Hz, (9)
3
where P
ℓ
is the transmit power of the ℓ-mode OAM beam.
III. THE EFFECT OF OBLIQUE ANGLE ON CHANNEL
CAPACITY
If the transmitter and receiver are aligned i.e. α = 0, the
channel matrix H is a circulant matrix that can be decomposed
into a diagonal matrix by one DFT matrix and one inverse
DFT (IDFT) matrix. Thus, the effective channel gain is equal
to the modulus of the ℓth eigenvalue of H, which is similar
to the eigenmode transmission in MIMO systems. However,
once α = 0, the circulant property of H does not hold any
longer. As a result, the OAM channel capacity deteriorates and
becomes worse than MIMO channel capacity. Therefore, it is
necessary to figure out the effect of oblique angle α on the
OAM channel capacity.
Assuming that the transmit and the receive UCAs are placed
in the far-field distance of each other [16], i.e. D ≫ R
t
and
D ≫ R
r
, thus we can approximate d
m,n
in (2) as
d
m,n
(a)
≈
R
2
r
+ R
2
t
+ D
2
−
R
r
R
t
cos φ cos θ
R
2
r
+ R
2
t
+ D
2
−
R
r
R
t
sin φ sin θ cos α − DR
r
sin θ sin α
R
2
r
+ R
2
t
+ D
2
(b)
≈ D −
R
r
R
t
D
cos φ cos θ −
R
r
R
t
D
sin φ sin θ cos α
+R
r
sin θ sin α, (10)
where (a) uses the method of completing the square and the
condition D ≫ R
t
, R
r
as same as the simple case
√
a
2
− 2b ≈
a −
b
a
, a ≫ b; (b) is directly obtained from the condition
D ≫ R
t
, R
r
. Then, substituting (10) into (1) and abbreviating
h(d
m,n
) to h
m,n
, we thus have
h
m,n
(c)
≈ β
λ
4πD
exp
− j
2π
λ
D + j
2πR
r
R
t
λD
cos φ cos θ
+ j
2πR
r
R
t
λD
+ sin φ sin θ cos α − j
2πR
r
λ
sin θ sin α
, (11)
where (c) neglects a few minor terms in the denominator of
the amplitude term and thus only 4πD is left. Having (11),
the modulus of the ℓ-mode effective OAM channel h
N
eff
(ℓ) is
derived as
|h
N
eff
(ℓ)| = |f (ℓ)Hf
H
(ℓ)|
=
1
N
N
n=1
N
m=1
h
m,n
exp
−j
2π(m − 1)l
N
+ j
2π(n − 1)l
N
≈
1
N
N
n=1
N
m=1
β
λ
4πD
exp
j
2πR
r
R
t
λD
cos
2π(n − m)
N
− j
2πD
λ
− j
2πR
r
α
λ
sin
2π(m − 1)
N
+ j
2π(n − m)l
N
q= n−m
= η
N−1
q=0
exp
j
2πR
r
R
t
λD
cos
2πq
N
+ j
2πql
N
·
N
m=1
exp
−j
2πR
r
α
λ
sin
2π(m − 1)
N
, (12)
0 2 4 6 8 10 12 14 16 18 20
oblique angle
°
0
0.5
1
1.5
2
|h
N
eff
(1)|
2
Conventional receiving w/o beam steering (Exact)
Conventional receiving w/o beam steering (Approx.(11))
Receiving w/ proposed beam steering (Exact)
Receiving w/ proposed beam steering (Approx.(16))
10
-2
(a)
0 0.5 1 1.5 2 2.5 3
oblique angle
°
0
0.2
0.4
0.6
0.8
1
1.2
1.4
|h
4
eff
(1)|
2
10
-8
Conventional receiving w/o beam steering (Exact)
Conventional receiving w/o beam steering (Approx.(12))
Receiving w/ proposed beam steering (Exact)
Receiving w/ proposed beam steering (Approx.(17))
(b)
Fig. 2: The modulus squared of the effective OAM channel
vs. oblique angle (a) large-scale UCAs; (b) 4-elements UCAs.
where η =
βλ
4π DN
. It is obvious to see from (12) that α is
only included in the second summation term, and if α = 0 all
the powers in the sum become 1 so that the sum achieves
its maximum value N. Otherwise, the modulus value of
the second summation term and thus the value of |h
N
eff
(ℓ)|
decreases rapidly even if α has a small value. The phenomenon
is easy to understand, because in the addition the N position
vectors with different arguments in the complex plane almost
cancel each other out when α >
λ
R
r
.
Let us examine the simple parameters N = 4 and ℓ = 1,
which are the same as the setting of the experimental system
in [6]. Then, |h
4
eff
(1)|
2
could be expressed as
|h
4
eff
(1)|
2
≈
λ
2
16π
2
D
2
sin
2πR
r
R
t
λD
+ cos
2πR
r
sin α
λ
sin
2πR
r
R
t
cos α
λD
2
. (13)
Thus, when α goes to zero, (13) could be further approximated
by the Taylor series expansion sin α ≈ α and cos α ≈ 1 −
α
2
2
,
and finally takes the form
|h
4
eff
(1)|
2
≈
λ
2
16π
2
D
2
sin
2
2πR
r
R
t
λD
1 + cos
2πR
r
α
λ
2
.
(14)
The (14) indicates that the value of |h
4
eff
(1)|
2
will fall from
its maximum at α = 0 to zero at α =
λ
2R
r
. Since in general
λ is much smaller than R
r
, the effective channel gain and
thus the OAM channel capacity drops rapidly as long as α
is comparable to
λ
2R
r
. We also illustrate the effect of oblique
angle on the effective channel gain with R
r
= 10λ, D =
1000λ, the large-scale UCAs (N = 5000) in Fig.2(a) and the
4
4-elements UCAs in Fig.2(b). The numerical results show that
the rapid decline occurs once α is beyond about 2 degrees.
IV. PROPOSED BEAM STEERING FOR THE MISALIGNED
UCA-BASED OAM SYSTEMS
To alleviate the performance degradation induced by the
misalignment, we propose applying the beam steering to the
UCA-based OAM communication systems, which is based on
the feasibility of tuning the angle of an OAM beam [12]. We
first deliberate on the beam steering in the non-parallel case
and then extend it to the off-axis and the general cases.
A. Non-parallel Case
For the non-parallel case as shown in Fig.3, the beam
steering approach is to compensate the changed phases caused
by oblique angle at the phase shifters of the receive UCA,
given that the direction of arrival (DOA) of the OAM beam
is perfectly estimated. Specifically, all the phase shifters can
be adjusted from the normal phases used for despiralization
only to the new phases used for both despiralization and beam
steering towards the DOA.
To develop the new phases, we reestablish the coordinate
system X’Y’Z’ with its origin located in the center of the
receive UCA as shown in Fig.1. Then, the azimuth and the
elevation angle of the incident OAM beam are respectively
−π/2 and α. Through calculating the phase difference be-
tween the reference element and the mth element of the receive
UCA, the receive beam steering vector a could be written as
a = [1, e
−jW
2
, ··· , e
−jW
N
], where
W
m
=
2πR
r
λ
cos
2π(m − 1)
N
+
π
2
sin α, (15)
m = 1, ··· , N. After involving these phases into the original
phases in f (ℓ) at the phase shifters of the receive UCA, the
ℓ-mode effective OAM channel becomes
h
′
N
eff
(ℓ) =
f(ℓ) ⊙ a
Hf
H
(ℓ). (16)
Thus, the modulus of the ℓ-mode effective OAM channel could
be written as
|h
′
N
eff
(ℓ)| ≈ η
N
m=1
N
n=1
exp
j
2π(n − 1)l
N
− j
2π(m − 1)l
N
+ j
2πR
r
R
t
λD
cos
2π(n − 1)
N
cos
2π(m − 1)
N
− j
2πD
λ
+ j
2πR
r
R
t
λD
sin
2π(n − 1)
N
sin
2π(m − 1)
N
cos α
(d)
≈ η
N
m=1
N−1
q=0
exp
j
2πR
r
R
t
λD
cos
2πq
N
+ j
2πql
N
· exp
−j
α
2
πR
r
R
t
λD
sin
2π(q + m − 1)
N
sin
2π(m − 1)
N
,
(17)
where (d) applies the substitution q = n − m and the
approximation cos α ≈ 1 −
α
2
2
with the focus on the small
values of α. Comparing (17) with (12), we find that in (17),
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Fig. 3: The diagram of the receive beam steering in the off-axis
misalignment case.
D appears in the denominator of the term containing α. Thus,
the effect of α is discounted greatly by D that is much larger
than R
t
, R
r
and α. It follows that the value of |h
′
N
eff
(ℓ)| is not
sensitive to the oblique angle
α
.
Corresponding to (13), we also examine the simple case that
N = 4 and ℓ = 1. The modulus squared of the effective OAM
channel with the receive beam steering can be expressed as
|h
′
4
eff
(1)|
2
≈
λ
2
4π
2
D
2
sin
2πR
r
R
t
λD
+ sin
2πR
r
R
t
cos α
λD
2
.
(18)
Similar to the explanation for (17), the effect of cos α is dis-
counted much by D. The mathematical analysis in (17), (18)
and the numerical results in Fig.2 indicates the effectiveness
of the receive beam steering approach.
B. Off-axis and General Cases
In the off-axis case as shown in Fig.4, the transmit UCA and
the receive UCA are parallel but not around the same axis. B is
the distance of axis deviation. Through introducing a virtual
UCA perpendicular to and in the middle of the connection
between the transmit and the receive UCA centers, the off-
axis case can be decomposed into two non-parallel cases: one
from the transmit UCA to the virtual UCA, and the other from
the virtual UCA to the receive UCA. Thus, the beam steering
in the off-axis case needs both of the transmit UCA and the
receive UCA steering their beams towards the virtual UCA.
Note that for the transmit UCA the azimuth and the el-
evation angle of the emergent OAM beam are π/2 and α
respectively. So, the transmit beam steering vector b could
be written as b = [1, e
−jW
′
2
, ··· , e
−jW
′
N
], where
W
′
n
=
2πR
t
λ
cos
2π(n − 1)
N
−
π
2
sin α, (19)
n = 1, ··· , N. Similarly, the ℓ-mode effective OAM channel
with both the transmit and the receive beam steering can be
formulated as
h
′′
N
eff
(ℓ) =
f(ℓ) ⊙ a
H
b
T
⊙ f
H
(ℓ)
. (20)
Only in this way can the performance degradation caused
by the off-axis misalignment be avoided to a large extent.
Furthermore, from geometrical model we know that the more
general misalignment cases could be solved by the combined
use of the transmit and the receive beam steering as well.
5
Fig. 4: The geometrical model of the transmit and the receive
UCAs in the off-axis misalignment case.
V. NUMERICAL RESULTS
In this section, to verify the beam steering method, we
simulate the UCA-based free space OAM communication
system, in which the number of antenna elements of UCA
N is equal to 12, the radius of the transmit UCA R
t
and that
of the receive UCA R
r
are both equal to 10λ. The transmit
SNR P
ℓ
/σ
2
n
is assumed to be 30dB.
In Fig.5 we compare the capacity of the OAM channel
without beam steering and the OAM channel with the proposed
beam steering in non-parallel and off-axis cases. When there
is the 2
◦
oblique angle, the OAM channel capacity decreases
to less than one tenth of its capacity in perfect alignment case.
When there is an axis deviation, the OAM channel capacity
also decays a large part and fluctuates with the distance of
array separation. The reason of the fluctuation is that with
the increase of array separation the fixed deviation distance
results in the varying oblique angle for the transmit and receive
beam steering. Fortunately, after applying the receive beam
steering in the non-parallel case and both the transmit and
the receive beam steering in the off-axis case, the misaligned
OAM channel capacity is almost the same as that in perfect
alignment case.
VI. CONCLUSION
In this paper, we investigate the effect of the misalignment
between the transmit and the receive UCAs and the beam
steering in the UCA-based free space OAM communication
system. We show that the effective OAM channel gain and
thus the OAM channel capacity drops rapidly as long as a
very small oblique angle exists. To deal with the problem,
we propose the transmit/receive beam steering approach to
circumvent the large performance degradation in the non-
parallel, the off-axis and other general misalignment cases,
which paves the way for the application of OAM in practice.
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