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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 67, NO. 11, NOVEMBER 2019
A Novel 2-D 3 × 3 Nolen Matrix for 2-D
Beamforming Applications
Han Ren , Student Member, IEEE, Hanxiang Zhang, Student Member, IEEE, Yuqi Jin,
Yixin Gu, Student Member, IEEE, and Bayaner Arigong , Member, IEEE
Abstract— In this paper, a 2-D beamforming phased array
using a novel 2-D Nolen matrix network is presented. The
Nolen matrix is a novel antenna feeding network composed of
only couplers with dedicated coupling ratios and phase shifters.
It does not require crossover and load termination compared
to other networks based on Butler and Blass matrix. To be
specific, the closed-form equations are derived first for uniplanar
single 3 × 3 Nolen matrix, which is composed of three couplers
and three phase delay lines. Most importantly, it is found that
the proposed Nolen matrix can employ couplers with arbitrary
phase differences to achieve relatively flexible progressive phase
delays across the radiating elements, presenting a high degree
of freedom on circuit topology and beamforming performance.
Then, a 2-D antenna feeding network is designed by stacking and
cascading six 3×3 Nolen matrices, and a 2-D patch antenna array
is integrated with the proposed feeding network to generate nine
radiation beams with unique directions on azimuth and elevation
planes, realizing the 2-D beamforming function. To verify the
proposed design concept, a prototype of 2-D beamforming phased
array operating at 5.8 GHz is designed, fabricated, and measured,
and the experimental results agree well with simulation and
theoretical analysis.
Index Terms— Antenna feeding network, beamforming network, flexible phase differences, microwave device, phased array.
I. I NTRODUCTION
N ANTENNA theory, phased array often means an array of
antennas with a feeding network. Through this microwave
network, each antenna element is fed by a signal with specific magnitude and phase, and the effective radiation pattern
generated by entire antenna array points toward a desired
direction, which is determined by the progressive phase delay
across the array elements. With emerging 5G technology and
Internet of Things, the intelligent software-defined wireless
network attracts great interest to connect a large number
of data-hungry mobile devices to minimize the hardware
constraints, and the advanced beamforming architecture is one
of the key technologies to boost the channel capacity and
increase the spectrum usage. In the beamforming antenna
I
Manuscript received February 3, 2019; revised April 17, 2019; accepted
April 29, 2019. Date of publication June 10, 2019; date of current version
November 5, 2019. (Corresponding author: Han Ren.)
H. Ren, H. Zhang, and B. Arigong are with the Department of Electrical Engineering, Washington State University, Vancouver, WA 98686 USA
(e-mail: hanren@my.unt.edu; bayaner.arigong@wsu.edu).
Y. Jin is with the Department of Mechanical and Energy Engineering,
University of North Texas, Denton, TX 76207 USA.
Y. Gu is with the Department of Electrical Engineering, University of Texas
at Arlington, Arlington, TX 76019 USA.
Color versions of one or more of the figures in this article are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMTT.2019.2917211
array, the feeding network is an important component to
allocate desired magnitude and phase delay of signal to each
array element, and the series-feed, parallel-feed, and matrix
feeding are three common topologies for implementing the
feeding network. Compared with the conventional series- and
parallel-feed networks, the matrix feeding network is a multiple input multiple output network and consists of components
such as couplers, phase shifters, crossovers, power dividers,
and switches. Among the matrix feeding networks, Butler
matrix [1]–[3] is a well-known type that features a symmetrical
structure with identical number of inputs and outputs and is
designed in a power of two scales such as in 4 × 4 scale
[4], 8 × 8 scale [5], and 16 × 16 scale [6]. Later, the Butler
matrix has been realized in unsymmetrical topologies, such as
2 × 4 scale [7], 3 × 4 scale [8], [9], 4 × 8 scale [10], [11].
Besides the circuit architecture, the Butler matrix has also
been studied in various ways: dual-band [12], [13], compact
size [14]–[16], broadband [17]–[23], low loss [24], sidelobe
control [25], flexibility [26], and beam steering [27], [28]. For
multibeam application, the Butler matrix has also been studied
to realize 2-D scanning function [29]–[33]. In [29], a uniplanar
beamforming network is designed by combining 4 × 4 Butler
matrix, hybrid coupler, and crossover to feed 2 × 4 antenna
array and generate 2-D scanning beams. With eight-port directional coupler and double-layer microstrip crossover, a 2-D
16 × 16 Butler matrix is implemented on a single substrate
board in [30]. In mm-wave frequency range, the substrate
integrated waveguide (SIW) has been applied in the design
of 2-D Butler matrix for low loss and easy integration in
various scales, such as 4 ×4 [31], 8 ×8 [32], and 16 ×16 [33].
The Blass matrix [34] is another type of matrix, which
has the same number of inputs and outputs as the Butler
matrix, with each input generating a unique phase difference
across its output ports. The main difference between the Butler
matrix and the Blass matrix is that the latter is composed of
couplers, phase shifters, and load terminations and removes
the crossovers. However, with structural limitation, part of the
signal in the Blass matrix flows into the terminated loads.
It turns out that the total efficiency is low compared to the
Butler matrix. Therefore, minimizing the power loss is the
most challenging issue for the Blass matrix, and there are very
few works on Blass matrix designs [35]–[38]. To overcome the
issue of the Blass matrix, the Nolen matrix [39] was designed
by cutting half of the Blass matrix along the diagonal line and
replacing the diagonal coupler by a transmission line. In such a
way, the Nolen matrix solves the power loss issue and reduces
0018-9480 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
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REN et al.: NOVEL 2-D 3 × 3 NOLEN MATRIX FOR 2-D BEAMFORMING APPLICATIONS
Fig. 1.
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Schematic of the proposed 2-D phased array including six 3 × 3 Nolen matrices and a nine-element antenna array for 2-D beamforming.
the number of couplers and phase shifters more than half to
achieve small form factor. Although the Nolen matrix has
great advantages, it has not been explored extensively and all
previous works were from one research group [40]–[42]. For
example, a microstrip line 4 × 4 Nolen matrix was designed at
S-band to match the same excitation laws of the conventional
4 × 4 Butler matrix [40], and the couplers applied in such
4×4 Nolen matrix must follow the strict requirement on phase
difference among output ports. In addition to that, a large number of phase shifters were employed inside and outside of the
Nolen matrix to compensate the phase misalignment. Another
example is the SIW 4 × 4 Nolen matrices [41], [42] presented
to improve the bandwidth and reduce the overall size.
In this paper, for the first time, a 2-D beamforming array
using 3 × 3 planar Nolen matrix is designed, fabricated, and
measured. Compared to previous research works, our proposed
3 × 3 Nolen matrix has the following features: 1) there is
no strict requirement for coupler’s phase difference. In other
words, the proposed Nolen matrix employs couplers with
arbitrary phase differences and generates relatively flexible
phase differences at its output ports; 2) it is constructed by
three couplers and three phase delay lines, and no additional
phase shifter is required outside of the matrix; and 3) the
phase delay of phase shifters applied in the proposed Nolen
matrix is flexible. This paper is organized as follows. First,
the closed-form equations are derived, and the detail design
approach is provided for the proposed 3 × 3 Nolen matrix
with flexible performance. Second, a 2-D beamforming array
is implemented by stacking and cascading six 3 × 3 Nolen
matrices and integrated with an antenna array. Third, the performance of the proposed design is verified in both simulation
and measurement, and all the results match well with the
theoretical prediction. This novel Nolen matrix could open up
new avenues for beamforming phased array designs in wireless
systems.
II. T HEORETICAL A NALYSIS
Our proposed 2-D phased array using novel 3 × 3 Nolen
matrix is shown in Fig. 1. Three 3 × 3 Nolen matrices
are stacked on the y-axis, while another three are piled up
on the x-axis. When one of nine input ports is excited,
equal magnitude and progressive phase delay on two axes
are generated at the output ports. After feeding these output
signals to a nine-element antenna array, a radiation beam is
obtained at a direction in 3-D space. Similarly, by switching
Fig. 2. (a) Schematic of 3×3 Nolen matrix. (b) Characteristics of a coupling
ratio C coupler.
the input ports, nine unique radiation beams will be generated
to achieve a significant 2-D beamforming function.
A. Design Theory for 3 × 3 Nolen Matrix
The proposed 3 × 3 Nolen matrix is a microwave network
with three input ports P1–P3 and three output ports P4–P6 as
shown in Fig. 2(a) and consists of three couplers and three
phase delay lines in a pyramidal layout. For couplers, one
with coupling ratio C1 is placed at the top of the pyramid,
while the other two with coupling ratio C2 are at the bottom.
Three transmission lines with 50- characteristic impedance
and electrical length θ1 , θ2 , and θ3 are used as phase delay
lines and connect the adjacent couplers. In general, a coupler
is a symmetrical four-port microwave device with two input
ports and two output ports. When the wave incidents from one
of two input ports, the signal is split into through and coupling
paths with specific magnitude and phase response. As shown
in Fig. 2(b), assuming: 1) the magnitude in the through path
is 1 − C, while the magnitude in the coupling path is C;
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 67, NO. 11, NOVEMBER 2019
2) the incident wave at port #1 generates phase shift α1 and
β1 in the through and coupling paths, and port #2 introduces
phase shift α2 and β2 in the two paths, respectively. In theory,
the coupler is a reciprocal, lossless, and energy conservation
network. By analyzing the scattering matrix of the coupler,
the phase shifts under different excitation ports follow the
condition as in [44]
(β1 − α1 ) + (β2 − α2 ) = π.
TABLE I
T WO N OLEN M ATRIX D ESIGN E XAMPLES W ITH C OUPLER C2 H AVING
T WO D IFFERENT VALUES OF P HASE D IFFERENCE C2
(1)
To be an antenna feeding network, the proposed 3×3 Nolen
matrix should achieve equal magnitude and three unique phase
differences at the output ports when the three inputs are excited
accordingly. To satisfy equal magnitude condition, the relation
between couplers with coupling ratios C1 and C2 is derived
as
1 − C1 = C1 · (1 − C2) = C1 · C2 = 1/3
(2)
where 1 − C1, C1·(1 − C2), and C1·C2 are the transmitting
coefficients from input port P1 to three output ports P4, P5,
and P6. As a result, C1 and C2 are equal to 2/3 and 1/2,
respectively. The scattering parameters of 3 × 3 network are
determined as
(3)
S41 = 1/3 · e j α1(C1)
j (α1(C2) +β1(C1) −θ3 )
S51 = 1/3 · e
(4)
j (β1(C1) +β1(C2) −θ3 )
(5)
S61 = 1/3 · e
j (α1(C2) +β2(C1) −θ1 )
S42 = 1/3 · e
(6)
S52 = 1/12 · e j (α2(C1) +2·α1(C2) −θ1 −θ3 )
+ 1/4 · e j (β1(C2)+β2(C2) −θ2 )
(7)
j (α2(C1) +α1(C2) +β1(C2) −θ1 −θ3 )
S62 = 1/12 · e
+ 1/4 · e j (α2(C2) +β1(C2) −θ2 )
(8)
j (β2(C1) +β2(C2) −θ1 )
(9)
S43 = 1/3 · e
j (α2(C1) +α1(C2) +β2(C2) −θ1 −θ3 )
S53 = 1/12 · e
+ 1/4 · e j (α2(C2) +β2(C2) −θ2 )
(10)
j (α2(C1) +β1(C2) +β2(C2) −θ1 −θ3 )
S63 = 1/12 · e
+ 1/4 · e j (2·α2(C2)−θ2 )
(11)
where Smn denotes the scattering parameter, and α1,2(C1)
and β1,2(C1) are phase delays in the through and coupling
paths of the coupler with coupling ratio C1 under wave
incidents at input ports #1 and #2, respectively. Similarly,
α1,2(C2) and β1,2(C2) represent the phase delays of through and
coupling paths in coupler with coupling ratio C2 when input
ports #1 and #2 are excited, respectively. Considering equal
magnitude |S41 | = |S51 | = |S61 | = |S42 | = |S52 | = |S62 | =
|S43 | = |S53 | = |S63 |, progressive phase distribution (S6n )
− (S5n ) = (S5n ) − (S4n ), and unique phase difference
(S61 ) − (S51 ) = (S62 ) − (S52 ) = (S63 ) − (S53 ) requirements, the electrical lengths of transmission lines connecting
adjacent couplers are derived as
θ2 − θ1 = β2(C2) − β2(C1) ∓ 90°
(12)
θ3 = C1 − C2 + α1(C2)
C1 = β1(C1) − α1(C1)
(13)
(14)
C2 = β1(C2) − α1(C2)
(15)
Fig. 3. (a) Schematic of coupler with coupling ratio 2/3. (b) Schematic
of coupler with coupling ratio 1/2. (c) Layout of the proposed 3 × 3 Nolen
matrix.
where C1 and C2 are phase differences of couplers with
coupling ratios C1 and C2, respectively. Substituting (12)–(15)
into (3)–(11), the phase differences among output ports of the
3 × 3 Nolen matrix are obtained as
1 = C2
(16)
2 = C2 ± 120°
3 = C2 ∓ 120°
(17)
(18)
where 1, 2, and 3 indicate the phase differences obtained
at its outputs when incident wave is applied at input ports P1,
P2, and P3, individually. It is obvious that phase differences
across the output ports of the proposed 3 × 3 Nolen matrix
depend on C2, which is the phase difference of the coupler
REN et al.: NOVEL 2-D 3 × 3 NOLEN MATRIX FOR 2-D BEAMFORMING APPLICATIONS
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Fig. 4. Simulation and measurement results of the proposed 3 × 3 Nolen matrix. (a)–(c) Transmission coefficient. (d)–(f) Phase difference. (g) Reflection
coefficient. (h) Isolation.
with coupling ratio C2. From (16) to (18), it is obvious that
3 − 2 = 2 − 1 = ±120°, which indicates that tuning
the phase difference of the coupler (here coupler with coupling
ratio C2) can achieve relatively flexible phase differences of
the 3 × 3 Nolen matrix. Two design examples summarized
in Table I exhibit this important characteristic of the proposed
idea. For design A, if the phase difference of the coupler with
coupling ratio C2 is −90°, at the outputs of the 3 × 3 Nolen
matrix, the phase differences between the adjacent output ports
are −90°, 150°, and 30° when the input ports P1, P2, and
P3 are excited in sequence. For design B where the coupler has
0° phase difference, the corresponding phase shifts between
adjacent ports are 0°, 120°, and −120° under different input
excitations. Based on (12)–(15), the proposed Nolen matrix
can use couplers with arbitrary phase differences C1 and
C2. Even given the fixed phase response of a coupler,
the electrical lengths of transmission lines θ1 and θ2 exhibit
relatively flexible values in (12). All these features make the
proposed design have a high degree of freedom on circuit
function, architecture, and couplers and phase delay lines.
Based on design A in Table I, a 3 × 3 Nolen matrix is
designed at 5.8 GHz. The schematic diagrams of the couplers
with the coupling ratio C1 = 2/3 and C2 = 1/2 are shown
in Fig. 3. The coupling ratio C1 coupler consists of two parallel
hybrid couplers and two open-end stubs as shown in Fig. 3(a)
[43]. The conventional quadrature hybrid coupler is applied for
coupling ratio C2 coupler in Fig. 3(b) [44]. Based on the phase
responses of the given couplers, the electrical lengths of the
transmission lines are calculated from (12) to (13), which are
θ3 = 90° and θ2 –θ1 = 35.26°. In this design, for the symmetric
purpose, θ1 is same as θ3 , and θ2 = 125.26°. The final circuit
for the proposed 3 × 3 Nolen matrix is shown in Fig. 3(c),
and the simulated S-parameter results are generated by ADS
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 67, NO. 11, NOVEMBER 2019
Fig. 6. (a) Perspective view of nine-element patch antenna array. (b) Patch
layer, ground layer, and cross section of a single-antenna element (top to
bottom).
Fig. 5. (a) 2-D antenna feeding network using six 3 × 3 Nolen matrices
with nine input ports 1–9. (b) 2-D beamforming phased array consisting of
six 3 × 3 Nolen matrices and a nine-element antenna array.
and shown in Fig. 4. At 5.8 GHz, the insertion loss is
within 4.9 ± 0.3 dB under different input excitations [shown
in Fig. 4(a)–(c)]. Within 1-dB variation of insertion loss,
the bandwidth of 3×3 Nolen matrix is 193, 231, and 311 MHz
under P1, P2, and P3 excitations, respectively. The phase
differences between the adjacent output ports under different
excitation sources are plotted in Fig. 4(d)–(f), where −90°,
150°, and 30° are obtained to match with the theoretical
values shown in Table I. The ±5° phase variation bandwidths
are 168, 451, and 141 MHz for three excitations. Fig. 4(g)
exhibits the return loss at input and output ports, and the
values are better than 24 dB at 5.8 GHz, where the 10-dB
bandwidth is 901 MHz (16%). The isolations between each
input and between each output ports are greater than 28 dB,
and the 10-dB bandwidth is 1.41 GHz [as shown in Fig. 4(h)].
All simulation results match well with the design theory to
demonstrate that the proposed 3 × 3 Nolen matrix realizes the
equal magnitude and three different progressive phase delays
across its output ports, which can be used in phased array
designs to achieve beamforming function.
B. Design of 2-D Phased Array for 2-D Beamforming
To achieve the 2-D beamforming, six of the proposed 3 × 3
Nolen matrices are stacked and cascaded to form a 2-D
antenna feeding network, and the 3-D view of the proposed
feeding network is shown in Fig. 5(a), where three Nolen
matrices are installed on the x-axis, while other three are on
the y-axis. There are nine input and nine output ports in this
feeding network, and the output ports feed 2-D patch antenna
array as shown in Fig. 5(b). To assemble the proposed 2-D
array, a mounting box and screws are designed to support
the whole circuit in 3-D printed housing. The detailed design
for nine-element patch antenna array working at 5.8 GHz is
shown in Fig. 6(a). The dimension of each patch is L =
14.68 mm and W = 27.12 mm, and the distance between
the adjacent elements is 33 mm. The probe feed is applied
from the backside, and a circle slot with 9.8 mm diameter
is made on the ground layer to optimize the antenna gain
[as shown in Fig. 6(b)]. The simulation shows the maximum
gain of the patch antenna is about 7.6 dBi. In theory, when
a uniform M × N planar antenna array is excited by incident
waves with identical magnitude for each element, the antenna
array factor (AF) is derived as
AF =
M
m=1
e j (m−1)(k·d x·sin θ cos φ+β x)
N
e j (n−1)(k·d y·sin θ sin φ+β y)
n=1
(19)
where k is the wavenumber, dx and dy are distances between
the adjacent antenna elements on x- and y-axes, βx and β y are
phase differences between the adjacent antenna elements on
x- and y-axes, and θ and are the radiation beam angles on
elevation and azimuth, respectively. From the antenna array
theory, it is known that the direction of the main radiation
beam is determined by the maximum value of the array factor.
Thus, the main beam radiation angle on elevation and azimuth
can be obtained from
k · d x · sin θ · cos φ + βx = 0
k · d y · sin θ · sin φ + βy = 0.
(20)
(21)
In Table II, the calculated phase differences of the proposed
2-D Nolen matrix feeding network on x-axis and y-axis are
listed in Columns 2 and 3, and Column 1 indicates that the
wave is applied on nine different input ports 1–9. The phase
shift on the x-axis is the phase difference of the adjacent
output ports of single 3 × 3 Nolen matrix stacked horizontally
REN et al.: NOVEL 2-D 3 × 3 NOLEN MATRIX FOR 2-D BEAMFORMING APPLICATIONS
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TABLE II
S UMMARY OF R ADIATION B EAMS ON E LEVATION A NGLE , A ZIMUTH A NGLE , AND G AIN U NDER I NCIDENT WAVE A PPLIED AT D IFFERENT I NPUT P ORTS
Fig. 7. Simulation results of the proposed 2-D beamforming phased array,
generating nine unique radiation beams with special values on elevation and
azimuth (θ , ) when input ports 1–9 are excited.
at the input side, and the phase shift on the y-axis indicates
the phase difference of adjacent outputs happened on 3 × 3
Nolen matrix stacked vertically at antenna side. Based on
the above-mentioned theoretical analysis, the main radiation
beams pointing toward nine unique directions in 3-D space
can be estimated from (20) to (21). To verify the theory,
full-wave 3-D electromagnetic simulation software HFSS is
applied to simulate the proposed 2-D beamforming phased
array in Fig. 5(b), and the simulation results are shown
in Fig. 7. It is clear that the nine radiation beams with nine
unique values (θ , ) match with the theoretical prediction,
and the total radiation gain is between 11.75 and 16.52 dBi.
The simulation demonstrates that our proposed 3 × 3 Nolen
matrix-based 2-D beamforming phased array contains a small
number of components and features a low power loss, which
will open a new direction for high figure of merit beamforming
phased array design. More than that, changing the phase
difference of the coupler in the proposed Nolen matrix will
Fig. 8. (a) Photograph of the fabricated 3×3 Nolen matrix. (b) Photograph of
the fabricated 2-D beamforming phased array. (c) Schematic of the experiment
setup for measuring 2-D beamforming.
generate flexible radiation beam angles in the 3-D dimension
as required.
III. E XPERIMENTAL R ESULTS
To verify the design concept, the proposed 3 × 3 Nolen
matrix working at 5.8 GHz is fabricated on Rogers RT/duroid
6002 laminate with the thickness of 0.508 mm, the loss
tangent of 0.0012, and the dielectric constant of 2.94 [as
shown in Fig. 8(a)]. The overall size is 71.39 × 31.61 mm2
(1.38λ × 0.61λ, λ is the wavelength of the operating frequency). The scattering parameters are measured using Agilent
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 67, NO. 11, NOVEMBER 2019
Fig. 9. Contour plots depicting measured nine radiation beam patterns generated by the proposed 2-D beamforming phased array on azimuth and elevation
angles.
PNA network analyzer, and the measurement results are shown
in Fig. 4. From the measurement, it is found that the center
frequency is shifted to 5.753 GHz due to the fabrication
tolerance, and Fig. 4(a)–(c) shows that the insertion loss under
different input port excitations is between 4.8 and 5.2 dB at
intersect frequency around 5.753 GHz. Within 1-dB variation,
the bandwidths of the fabricated Nolen matrix are 207, 229,
and 280 MHz for excitations at input ports P1–P3, respectively. The phase differences between adjacent output ports
under different input excitations are plotted in Fig. 4(d)–(f).
At 5.753 GHz, it is clear to find that the phase differences
are −90°, 150°, and 30°, matching well with the theoretical
analysis. Within ±5° tolerance range, the bandwidths for different input excitations are 130, 255, and 117 MHz. Fig. 4(g)
exhibits the return loss at input and output ports, and they
are better than 16 dB at the center frequency. The 10-dB
bandwidth for all ports is about 439 MHz (8%). The isolations
between each input and each output ports are greater than
18 dB, and the 10-dB bandwidth is 1.49 GHz [as shown
in Fig. 4(h)]. Overall, all measurement results agree well with
the theoretical analysis and simulation results to further prove
that the proposed 3 × 3 Nolen matrix splits the signal with
equal magnitude and achieves three unique phase differences
at its outputs.
To experimentally verify the 2-D beamforming phased array,
the fabricated six 3 × 3 Nolen matrices are assembled in
a mounting box as shown in Fig. 8(b), which is designed
and built by a 3-D printer using acrylonitrile butadiene
styrene (ABS). All ports in Nolen matrix and antenna array
TABLE III
C OMPARISON B ETWEEN THE P ROPOSED 3 × 3 N OLEN M ATRIX
W ITH O THER M ATRIX D ESIGNS
are soldered with SMA connectors. RF adapters and coaxial
cables are applied to connect all Nolen matrices and patch
antenna array together. For the radiation pattern measurement,
the mounting box is installed on a positioner and rotated in
full spherical range. A calibrated broadband horn antenna
is applied as receiver, and Agilent PNA network analyzer
is used to measure the S-parameters [shown in Fig. 8(c)].
By sequentially exciting nine input ports, the nine radiation
beams in different directions are plotted as a contour on
azimuth over elevation angles using DAMS software and
REN et al.: NOVEL 2-D 3 × 3 NOLEN MATRIX FOR 2-D BEAMFORMING APPLICATIONS
Fig. 10.
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(a)–(i) Simulation and measurement results of normalized E- and H-plane patterns with input ports 1–9, respectively.
shown in Fig. 9, and the measured data are summarized
in Table II. Since nine input ports contribute nine groups of
phase differences on the x-axis and the y-axis (as shown in
Columns 2 and 3 in Table II), nine corresponding radiation
beam angles on azimuth and elevation are measured (as shown
in Columns 6 and 7 in Table II). Compared with the simulation results in Columns 4 and 5 in Table II, the maximum
imbalance from measurement on elevation and azimuth is 8°
and 25°, respectively, which are caused by the tolerance from
fabrications of six Nolen matrix boards and radiation patterns
measurement in open space. The measured gains of radiation
beams are between 9.43 and 19.2 dBi shown in Column 8
in Table II. The maximum gain is obtained when input port
P9 is excited, which has the lowest sidelobe level (SLL) due to
the smallest phase difference on the x- and y-axes. In contrast,
the maximum phase differences on two axes are generated
under input port P5 excitation, causing gain reduction because
of the worst case SLL. Fig. 10 shows the simulation and
measurement of normalized E-/H-plane patterns with inputs
1–9, and the main beam angle in each pattern agrees well
with the results in Fig. 9. In E-plane (elevation), the measured
maximum half-power beamwidth (HPBW) is 34°, while the
minimum is 8°. In the simulation, the HPBW in E-plane is
within 26°–33°. For H-plane (azimuth), the measured and simulated HPBWs are within 13°–79° and 31°–162°, respectively.
In Fig. 10(i), the maximum gain (as shown in Table II) is
obtained with input port P9 excitation, and it is caused by a
few sidelobes in both planes. The H-plane pattern of input
port P5 excitation as in Fig. 10(e) has five obvious sidelobes
in which the SLL is less than 10 dB. SLL control for feeding
network has been extensively studied in previous works, and
the typical methodologies are summarized as optimizing the
interantenna space, nonuniform tapered magnitude distribution, and dynamic input power compensation. In general, both
simulation and measurement results match well with each
other to verify the performance of the 2-D 3 × 3 Nolen matrix
for 2-D beamforming. The comparison with other matrix
designs is summarized in Table III. It is obvious that our
proposed 3 ×3 Nolen matrix requires the minimum number of
components, removes the crossovers, generates flexible output
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 67, NO. 11, NOVEMBER 2019
phase differences, and adopts arbitrary couplers. Although
three phase differences are generated in the proposed 3 × 3
Nolen matrix, the flexible phase differences take over the
disadvantage of total beam number in real applications. Compared with the other 3 × 3 Butler matrix designs [45], [46],
although the couplers have the same coupling ratios as our
design, the proposed 3 × 3 Nolen matrix can generate progressive phase distribution for the linear antenna array. From
S-parameters comparison, our 3 × 3 Nolen matrix achieves the
best magnitude imbalance and obtains reasonable values on
phase imbalance and return loss. Comparing the bandwidth,
4 × 4 [26] and 3 × 3 [46] Butler matrices are better than
the proposed Nolen matrix. However, our design still provides
wider bandwidth compared to some Butler matrix [8], [9]
and Nolen matrix [36]. To further improve the bandwidth,
the couplers and phase shifters in the proposed design can
be replaced by broadband couplers and phase shifters using
stripline structure.
IV. C ONCLUSION
A novel beamforming network based on 3 × 3 Nolen matrix
has been demonstrated to realize the equal magnitude and
three different phase differences between adjacent outputs by
exciting three input ports. In addition, a 2-D phased array
based on the proposed 3 × 3 Nolen matrix is designed and
characterized for 2-D beamforming, and nine unique radiation
beams are measured with reasonable gains. The proposed
novel design can provide a new approach for next-generation
5G multiple-in multiple-out (MIMO) phased array designs
and pave the way toward developing beamforming systems
with high figure of merit. In our next work, a fully tunable
3 × 3 Nolen matrix will be investigated to achieve full
phase differences ranging from −180° to 180° to build beam
steering capability in 5G mm-wave wireless communication
systems.
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Han Ren (S’12) was born in Nanjing, China.
He received the B.S. degree in electrical engineering from the Nanjing University of Posts and
Telecommunications, Nanjing, in 2008, and the M.S.
and Ph.D. degrees in electrical engineering from
the University of North Texas, Denton, TX, USA,
in 2013 and 2017, respectively.
He is currently a Post-Doctoral Researcher in electrical engineering with Washington State University,
Vancouver, WA, USA. His current research interests
include RF/microwave active and passive circuits,
phased array antenna, and metasurface/metamaterial.
4631
Hanxiang Zhang (S’18) was born in Yangzhou,
China. He received the B.S. degree in electrical
engineering from Huaiyin Normal University,
Huaian, China, in 2016, and the M.S. degree
in electrical engineering from Washington State
University, Vancouver, WA, USA, in 2019.
His
current
research
interests
include
RF/microwave circuit and system design.
Yuqi Jin was born in Shanghai, China. He received
the B.S. degree in mechanical and energy engineering from the University of North Texas, Denton, TX,
USA, in 2016, where he is currently pursuing the
Ph.D. degree.
His current research interests include functional
phononic crystals, acoustic metamaterials, and ultrasound elastographic imaging.
Yixin Gu (S’17) was born in Wuxi, China.
He received the B.S. degree in computer science
from Jiangnan University, Wuxi, in 2004, and the
M.S and Ph.D. degrees in electrical engineering from
the University of North Texas, Denton, TX, USA,
in 2014 and 2018, respectively.
His current research interests include embedded
system and wireless communication.
Bayaner Arigong (M’08) was born in Ordos, China.
He received the B.Sc. and M.Sc. degrees from the
China University of Geosciences (CUG), Wuhan,
China, in 2005 and 2008, respectively, and Ph.D.
degree in computer science and engineering from
the University of North Texas, Denton, TX, USA,
in 2015.
From 2015 to 2017, he was an Advanced RF
System Design Engineer with Infineon Technologies, where he was involved in developing highperformance integrated power amplifier circuit for
cellular base stations. Since 2017, he has an Assistant Professor with the
Electrical Engineering Department, Washington State University (WSU),
Vancouver, WA, USA. His current research interests include RF/microwave
circuits and systems (e.g., passive circuit, beamforming architecture, power
amplifiers, antenna, phased array, and RF front end), metamaterials, transformation optics, and nanophotonics.