Nolen Matrix Research Articles

A tunable 3×5 Nolen matrix for sidelobe-reduced and beamforming applications

A methodology of designing a 3×5 Nolen matrix with non-uniform power distribution is proposed in this paper. Several tunable phase shifters are designed to realize the capability of continuous beam steering. Additionally, two power dividers are integrated with a 3×5 Nolen matrix to produce the desired non-uniform power distribution, thereby reducing the sidelobe levels. In order to establish the appropriate power divider ratios, this paper thoroughly examines the correlation between power ratio at each port and different loss scenarios as well as Chebyshev distribution. An experimental Nolen matrix prototype has been designed and fabricated to confirm the efficacy of our proposed approach. The fabricated 3×5 Nolen matrix prototype offers continuous beam steering and achieves sidelobe levels below −22 dB.

Tunable 3 × 3 Nolen Matrix Network for Power-Saving Phased Array

Tunable 3 × 3 Nolen Matrix Network for Power-Saving Phased Array

Design and implementation of a multibeam filtering antenna array using compact 3 × 3 Nolen matrix aided by grey wolf algorithm

This paper presents a novel multibeam filtering antenna array based on a compact 3 × 3 Nolen matrix integrated with HRR filters for solar observations. The design facilitates the simultaneous recording of solar radio flux density and sky background flux density, streamlining calibration processes. The use of stub-loaded transmission line (SLTL) couplers achieves a significant 43.4% reduction in the size of the Nolen matrix, enhancing compactness. The hairpin ring resonator (HRR) filter with two transmission zeros (TZs) is employed to ensure electromagnetic compatibility (EMC). Given the limitations of conventional manual design methods, the intelligent grey wolf algorithm (GWA) is adopted to optimize the physical parameters of the Nolen matrix layout, realizing rapid and accurate achievement of the desired performance. The −6.2±0.15 dB transmission coefficient and 0°, −120°and +120°of output phase differences within 2° phase error are accomplished at 2.8 GHz. Finally, the proposed multibeam filtering antenna array is fabricated and measured to confirm its ability to generate the required three beams with filtering characteristics. The measured results have a good agreement with the simulated results.

Design of Extended Nolen Matrix With Enhanced Beam Controllability and Widened Spatial Coverage

Design of Extended Nolen Matrix With Enhanced Beam Controllability and Widened Spatial Coverage

High-Capacity Multiple-Input Multiple-Output Communication for Internet-of-Things Applications Using 3D Steering Nolen Beamforming Array

In this paper, a novel 2D Nolen beamforming phased array with 3D scanning capability to achieve high channel capacity is presented for multiple-input multiple-output (MIMO) Internet-of-Things (IoT) applications. The proposed 2D beamforming phased array is designed by stacking a fundamental building block consisting of a 3 × 3 tunable Nolen matrix, which applies a small number of phase shifters with a small tunning range and reduces the complexity of the beam-steering control mechanism. Each 3 × 3 tunable Nolen matrix can achieve a full 360° range of progressive phase delay by exciting all three input ports, and nine individual radiation beams can be generated and continuously steered on azimuth and elevation planes by stacking up three tunable Nolen matrix in horizontal and three in vertical to maximize signal-to-noise ratio (SNR) in the corresponding spatial directions. To validate the proposed design, the simulations have been conducted on the circuit network and assessed in a fading channel environment. The simulation results agree well with the theoretical analysis, which demonstrates the capability of the proposed 2D Nolen beamforming phased array to realize high channel capacity in MIMO-enabled IoT communications.

Control-Relaxed Wideband 2×4 Nolen Matrix With 360° Continuously Tuned Differential Phase

Control-Relaxed Wideband 2×4 Nolen Matrix With 360° Continuously Tuned Differential Phase

Compact Wideband 3 × 3 Nolen Matrix With Couplers Integrated With Phase Shifters

Compact Wideband 3 × 3 Nolen Matrix With Couplers Integrated With Phase Shifters

Wideband 4×4 Nolen Matrix with 360° Continuously Tuned Differential Phase and Low In-Band Phase Deviation Error

In the paper, a wideband tunable 4×4 Nolen matrix is proposed. By using the presented topology of the Nolen matrix, tunable output port phase differences can be realized by inserting one type of tunable phase shifters (T-PS). Besides, the in-band phase deviation error is minimized by adding compensation phase shifters (C-PSs) to the output ports of the Nolen matrix. Analytical design methods based on signal flow graphs and complex exponential signal are given to obtain the rigorous relationships of the coupling and phase shifts in the Nolen matrix. Besides, a detailed method for output ports phase slope compensation is provided. For validation, a prototype centered at 5.8 GHz is designed and fabricated. Measurement results agree well with the simulated ones. By using only one voltage to control the phase shifts of the T-PS in a 90° range for each input excitation of the Nolen matrix, a full 360° range of the progressive phase difference is realized by switching four input ports. The measured fractional bandwidth under 10 dB return loss and isolation is larger than 24.5% with the in−band±1.5 dB amplitude imbalance and ±15° phase deviation error for ports 1-4 excitations. Besides, for a more strict criterion of ±1 dB amplitude imbalance and 10° phase deviation error, the measured bandwidths are larger than 15% for all port excitations.

Dimension Expansion Method and Implementation of Broadband Planar 2-D 3 × 3 Nolen Matrix

Dimension Expansion Method and Implementation of Broadband Planar 2-D 3 × 3 Nolen Matrix

Beamforming networks using the Nolen matrix for 5G applications

A design of a wideband low loss 3 × 3 Nolen matrix is presented at 3.5 GHz. The proposed Nolen matrix is designed with the implementation of three branch-line couplers and two phase shifters. The interdigital capacitor unit cell is implemented in both the coupler and phase shifter to reduce the size and improve the bandwidth. The proposed 3 × 3 Nolen matrix is simulated using computer simulation technology (CST) software and then fabricated on the FR4 substrate with . The 3 × 3 Nolen matrix performance is achieved with a fractional bandwidth of 49% and operates at 2.52–4.2 GHz. The coupling factor of the 3 × 3 Nolen matrix across outputs at 3.5 GHz is −6 ± 0.98 dB with phase differences of 3.37° at port 1, 125.6° at port 2, and −117.8° at port 3. Then, the proposed Nolen matrix is attached with three standard microstrip antennas to prove the property of a unique beam scanning. The measured results approve the concept of three different phases scanning at 0°, 45°, and −40° with a reflection coefficient below −10 dB. Compared to the conventional Nolen matrix, the 3 × 3 Nolen matrix with a modified coupler is achieved with a low loss characteristics with phase error up to 2.2° and a compact size of 0.82 λ × 0.75 λ. The proposed Nolen matrix is suitable to be used in future 5G antenna beamforming applications.

A General Approach for Synthesizing Multibeam Antenna Arrays Employing Generalized Joined Coupler Matrix

Despite the rapidly increasing interest in analog multibeam antennas, there has been a lack of systematic theoretical approaches to synthesizing circuit-type multiple beamforming networks, such as the Blass matrix and the Nolen matrix. To address the issue, this article presents a new concept, the generalized joined coupler (GJC) matrix, which encapsulates both the Blass matrix and the Nolen matrix, as well as their variants, and presents a novel theoretical framework for generating individually and independently controllable multiple beams using the GJC matrix. A GJC matrix has <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N $ </tex-math></inline-formula> columns to feed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N $ </tex-math></inline-formula> antenna elements and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M $ </tex-math></inline-formula> rows to feed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M $ </tex-math></inline-formula> beams, and the direction of each individual beam can be controlled by tuning the phase shifters in the associated row of the GJC matrix. In this article, a matrix theory is developed, and an optimization algorithm is proposed to provide a mathematical tool for synthesizing such matrices and, consequently, the multiple beams. Using a particle swarm optimization algorithm, numerical results demonstrate that multibeams with independent control of individual beam directions and sidelobes can, indeed, be synthesized in a systematic manner. Specifically, two GJC matrix variants, the Blass-like matrix and the Nolen-like matrix, are investigated.

Wideband Hybrid Couplers With Unequal Power Division/Arbitrary Output Phases and Applications to Miniaturized Nolen Matrices

In this article, a new design approach for developing hybrid couplers with arbitrary power division and the phase difference is presented. The design is based on a transformer-based coupler with high isolation between sum and difference ports. Based on this model, a modified coupler is developed by including two L-shaped networks to produce constant phase differences at two output ports. Slotlines and microstrip-to-slotline transitions are used for expanding the bandwidth and introducing a 180° phase difference at the difference port. Two prototypes with equal and unequal power division are designed and verified. Experimental results reveal that the proposed designs can achieve constant phase difference, low insertion loss, and satisfactory matching across a broad bandwidth of more than 40%. Based on the proposed designs, a simplified multibeam Nolen matrix is also designed utilizing only three hybrid couplers and no phase shifters. The compact beamforming network demonstrates that the proposed component is very useful for size reduction in wireless communication systems.

Design of wideband 4 × 4 Nolen matrix using m ultistub loaded phase shifters

In this article, a wideband microstrip 4 × 4 Nolen matrix using novel phase shifters is proposed. Because of the unequal electrical paths connecting one input to every output, Nolen matrices in standard form usually introduce significant phase dispersion and amplitude instability, limiting the effective bandwidth. A novel phase shifter consisting of multiple open- and short-circuited stubs is designed, whose rate of change of phase with frequency can be conveniently adjusted. This phase shifter can be used to balance the rate of change of phase with frequency of the unequal electrical paths to reduce the phase dispersion and amplitude instability, thus realizing wide bandwidth of the output phase differences and output amplitudes. To verify the proposed concept, a wideband 4 × 4 Nolen matrix operating at 6 GHz is designed and fabricated, showing a bandwidth up to 33% within ±15° phase imbalance and 21.5% within ±1 dB amplitude imbalance.

Analytical Design Method and Implementation of Broadband 4 × 4 Nolen Matrix

Beamforming techniques are widely used in antenna array systems. One of the most popular beamforming networks (BFNs) is Nolen matrix, which can achieve flexible phase progression. However, due to the lack of analysis on phase slope alignments between internal components, the conventional topology-based Nolen matrix exhibits narrow bandwidth. To overcome this shortage, an analytical design method based on signal flow graphs (SFGs) and complex exponential signal is proposed. With the use of this analytical design method, rigorous relationships of phase slope alignments between couplers and phase shifters within the operating band in the traditional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4\times $ </tex-math></inline-formula> 4 Nolen matrix topology can be derived. Based on this, a novel <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4\times $ </tex-math></inline-formula> 4 Nolen matrix topology composed of couplers and differential phase shifters (DPSs) with specific phase difference slopes is proposed. For verification, a broadband <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4\times $ </tex-math></inline-formula> 4 Nolen matrix based on the proposed topology centered at 1.8 GHz, is designed, fabricated, and measured. Simulated and measured results are in good agreement. The measured bandwidths of ports 1–4 defined by return losses and isolations (15 dB for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{i1}$ </tex-math></inline-formula> , 10 dB for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{ij,i = 1-4, j = 2-4}$ </tex-math></inline-formula> ) and 1 dB amplitude imbalance are 41.49%, 46.21%, 40%, and 48.24%, respectively. Phase errors are within ±5° for 0° and ±90° phase progressions and vary within the range from −10° to 5° for 180° phase progression. The proposed Nolen matrix exhibits the widest bandwidth and the flattest phase progression among Nolen matrices in existence, which verifies the validity of the proposed analytical design method.

Generalized One-Dimensional Parallel Switching Matrices With an Arbitrary Number of Beams

This paper introduces a novel family of one-dimensional switching matrices, also known as beamforming networks (BFNs), which can be sized for an arbitrary number of beams. It also describes the associated numerical design method, supported by several design examples. The concept is further validated with a full-wave model of a generic 5 × 5 switching matrix designed at 76 GHz. The proposed matrices are obtained by cascading units consisting of a two-way coupler and a phase shifter in a parallel configuration, providing theoretically lossless operation as a direct consequence of the orthogonal BFN topology and equalized path lengths, expected to benefit the frequency bandwidth. The design method is general, also including the configurations equivalent to Butler matrices when the number of beams is a power of two. The obtained planar matrix designs combine some benefits of the well-known Butler and Nolen matrices, namely the parallel topology and the arbitrary number of beams. The minimum number of cascaded units required by the proposed topology, also corresponding to the number of layers, is found to be <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> is the number of beams, which is less than the number of cascaded units for the corresponding “parallel” Nolen matrix, for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> larger than three, thus providing a reduction in the number of layers of up to 38% for an 8-beam matrix. This reduction is close to the theoretical upper boundary of 50%, reached for large values of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> . This significant reduction is expected to have a positive impact on RF performance, and in particular insertion losses, as well as amplitude and phase dispersion over the operating band. Interestingly, the proposed switching matrix configuration reduces to a standard Butler matrix when the number of beams is equal to 4, and demonstrates a reduction in the number of layers of 20% over the corresponding planar Butler matrix for a design with 8 beams. The proposed matrix configuration is a promising solution for the design of low-cost single-layer beam-switching matrices.

A Symmetric Beam-Phased Array Fed by a Nolen Matrix Using 180° Couplers

This letter presents a new microstrip line-phased array using $3 \times 3$ Nolen matrix – with three inputs and three output ports. For each input port excitation, the signal is divided into three equal magnitudes and unique progressive phase shifts at its output ports. The proposed microwave network is composed of three 180° couplers with two of them having identical coupling factors and two microstrip line-phase shifters. To verify the proposed design concept, a prototype of the feeding network working at 5.8 GHz is designed, fabricated, and measured. Both simulation and measurement results are in good agreement with the theoretical prediction.

Ultra‐compact 3 × 3 Nolen matrix beamforming network

A novel compact 3 × 3 Nolen matrix beamforming network is proposed in this study. Unlike the Butler matrix and Blass matrix, in this design, the proposed feeding network only requires couplers and phase shifters and removes crossovers and load terminations. To minimise the size, the authors’ proposed novel Nolen matrix is realised by lumped-element couplers. First, the design theory for 3 × 3 Nolen matrix is derived, and the associated pyramidal layout consisting of three couplers with two coupling ratios and three-phase shifters is proposed. Based on the determined topology, the S-parameters of the proposed feeding network and desired coupling ratios are derived, and it is found that the phase differences among the output ports are dependent on the characteristics of the couplers. Then, each coupler is realised by using lumped-elements (including inductors and capacitors) in the proposed topologies, where the values of the elements are obtained by analysing the ABCD-matrix of the branch lines in the couplers. To verify the proposed concept, an experimental prototype working at 1 GHz is fabricated and tested. The simulation and measurement results agree well with each other, and an ultra-compact size 0.17λ × 0.06λ is implemented to achieve more than 90% size reduction comparing with the previous designs.

A Novel 2-D $3\times3$ Nolen Matrix for 2-D Beamforming Applications

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 $3 \times 3$ 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 $3 \times 3$ 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.

26 GHz phase shifters for multi-beam nolen matrix towards fifth generation (5G) technology

This paper presents the designs of phase shifters for multi-beam Nolen matrix towards the fifth generation (5G) technology at 26 GHz. The low-cost, lightweight and compact size 0° and 45° loaded stubs and chamfered 90°, 135° and 180° Schiffman phase shifters are proposed at 26 GHz. An edge at a corner of the 50 Ω microstrip line Schiffman phase shifter is chamfered to reduce the excess capacitance and unwanted reflection. However, the Schiffman phase shifter topology is not relevant to be applied for the phase shifter less than 45° as it needs very small arc bending at 26 GHz. The stubs are loaded to the phase shifter in order to obtain electrical lengths, which are less than 45°. The proposed phase shifters provide return loss better than 10 dB, insertion loss of -0.97 dB and phase difference imbalance of ± 4.04° between 25.75GHz and 26.25 GHz. The Rogers RT/duroid 5880 substrate with dielectric constant of 2.2 and substrate thickness of 0.254 mm is implemented in the designs.

NOVEL NOLEN MATRIX BASED BEAMFORMING NETWORKS FOR SERIES-FED LOW SLL MULTIBEAM ANTENNAS

A new type of cascaded series feed beamforming networks (BFNs) is introduced. The network architecture is based on a dual- series Nolen matrix topology. It is able to produce tapered output amplitude distributions from N £ N conflgurations. The general concept, analysis and systematic design of the BFNs are given. The networks are designed and intended to be used mainly for low Sidelobe Level (SLL) linear Multibeam Antennas (MBAs). Several design examples are presented, along with fabrication and measurements of an S-band prototype.