Beampattern Design Research Articles

On the Design of Target Beampatterns for Differential Microphone Arrays

Differential microphone arrays (DMAs) have many interesting properties and have been widely used in acoustic, audio, and speech applications. A critical part of a DMA is the differential beamformer, which is generally designed in two important steps: 1) specifying a target beampattern based on what differential sound pressure field the DMA is expected to respond to and 2) designing the differential beamforming filter so that the resulting beampattern matches the target one. Most efforts in the study of DMAs so far have focused on the second step while choosing one of the limited patterns available in the literature as the target beampattern. Since it governs how the array performs, how to design the target beampattern is an important problem, which this paper addresses. The major contributions of this paper consists of the following four aspects. First, a positive superposition theorem is presented, which shows that the linear combination of effective beampatterns with non-negative coefficients is always an effective beampattern. Second, we propose a general approach to the design of target DMA beampatterns based on the positive superposition theorem. Third, an overview of the classical target beampatterns is provided and discussion is made on how to form effective base patterns. Fourth, we show that the smallest first null of a DMA is π(2N) with N being the DMA order, which provides the rule of setting nulls in practice. Finally, with examples, we show that with the use of the alternating-direction-method-of-multipliers algorithm, the proposed approach is able to generate useful DMA target beampatterns.

Waveform covariance matrix design using Fourier series coefficients

Multiple-input multiple-output (MIMO) radars may outperform other radar systems such as phased array radars, in terms of higher resolution, better detection probability in the presence of interferences, better parameter identifiability and more flexibility in beampattern design. Waveform covariance matrix design, because of its role in the beampattern synthesis process, is one of the most important problems in MIMO radar systems. In this study, the authors have proposed a closed-form solution based on Fourier series coefficients to design a covariance matrix. The resulting covariance matrix fulfils the practical constraints, i.e. positive semi-definiteness and the uniform elemental power constraint. It also provides performance similar to that of iterative methods, while requires lower computation time and provides better mean square error with respect to other existing closed-form methods. Eigenvalue decomposition is also utilised to convert the possible resulted pseudo-covariance matrices (pseudo-CM), which are not guaranteed to be positive semidefinite, into a covariance matrix. Simulation results show the performance of the proposed method.

Transmit MIMO Radar Beampattern Design via Optimization on the Complex Circle Manifold

The ability of Multiple-Input Multiple-Output (MIMO) radar systems to adapt waveforms across antennas allows flexibility in the transmit beampattern design. In cognitive radar, a popular cost function is to minimize the deviation against an idealized beampattern (which is arrived at with knowledge of the environment). The optimization of the transmit beampattern becomes particularly challenging in the presence of practical constraints on the transmit waveform. One of the hardest of such constraints is the non-convex constant modulus constraint, which has been the subject of much recent work. In a departure from most existing approaches, we develop a solution that involves direct optimization over the non-convex complex circle manifold. That is, we derive a new projection, descent, and retraction (PDR) update strategy that allows for monotonic cost function improvement while maintaining feasibility over the complex circle manifold (constant modulus set). For quadratic cost functions (as is the case with beampattern deviation), we provide analytical guarantees of monotonic cost function improvement along with proof of convergence to a local minima. We evaluate the proposed PDR algorithm against other candidate MIMO beampattern design methods and show that PDR can outperform competing wideband beampattern design methods while being computationally less expensive. Finally, orthogonality across antennas is incorporated in the PDR framework by adding a penalty term to the beampattern cost function. Enabled by orthogonal waveforms, robustness to target direction mismatch is also demonstrated.

Spatio-Spectral Radar Beampattern Design for Coexistence With Wireless Communication Systems

We address the problem of designing a transmit beampattern for multiple-input multiple-output (MIMO) radar considering coexistence with wireless communication systems. The designed beampattern is able to manage the transmit energy in spatial directions as well as in spectral frequency bands of interest by minimizing the deviation of the designed beampattern versus a desired one under a spectral constraint as well as the constant modulus constraint. While unconstrained beampattern design is straightforward, a key open challenge is jointly enforcing the spectral constraint in addition to the constant modulus constraint on the radar waveform. A new approach is proposed in this paper, which involves solving a sequence of constrained quadratic programs such that constant modulus is achieved at convergence. Furthermore, we show that each problem in the sequence has a closed form solution leading to analytical tractability. We evaluate the proposed beampattern with interference control (BIC) algorithm against the state-of-the-art MIMO beampattern design techniques and show that BIC achieves closeness to an idealized beampattern along with desired spectral shaping.

Transmit beampattern design for coherent FDA by piecewise LFM waveform

In transmit beampattern design, it is desirable to maximize the power around locations of interest. Multiple-input multiple-output (MIMO) radar transmits orthogonal waveforms, which is hardly achievable in practical applications. Moreover, the constant modulus constraint and the computational burden limit the system performance. In this paper, we propose an effective piecewise linear frequency modulation (PLFM) waveform for coherent frequency diverse array (FDA) radar to synthesize the desired transmit beampattern. By deriving the relationship between the beam direction and the transmit signal frequency, we find that different sub-band of transmitted signal corresponds to different angular sector. With the relationship between frequency band and mainlobe of beampattern, the beampattern synthesis is transformed into the design of waveform spectrum which is realized by designing the PLFM waveform in this paper. Compared with the state-of-the-art technique, the proposed PLFM waveform in coherent FDA enjoys the advantage of the constant modulus, low sidelobe level and reduced computation complexity. In addition, the multi-dimensional ambiguity function and the transmit beampattern response are derived to examine the properties. Numerical results are presented to validate the effectiveness of the proposed method.

Insights Into Frequency-Invariant Beamforming With Concentric Circular Microphone Arrays

This paper studies the problem of frequency-invariant beamforming with concentric circular microphone arrays (CCMAs) and presents an approach to the design of frequency-invariant and symmetric beampatterns. We first apply the Jacobi-Anger expansion to each ring of the CCMA to approximate the beampattern. The beamformer is then designed by using all the expansions from different rings. In comparison with the existing work in the literature where a Jacobi-Anger expansion of the same order is applied to different rings, here in this contribution the order of the Jacobi-Anger expansion at a ring is related to its number of sensors and, as a result, the expansion order at different rings may be different. The developed approach is rather general. It is not only able to mitigate the deep nulls problem in the directivity factor and the white noise gain, that is common to circular microphone arrays (CMAs), and improve the steering flexibility, but is also flexible to use in practice where a smaller ring can have less microphones than a larger one. We discuss the conditions for the design of $N$ th-order symmetric beampatterns and examples of frequency-invariant beampatterns with commonly used array geometries such as CMAs, CMAs with a sensor at the center, and CCMAs. We show the advantage of adding one microphone at the center of either a CMA or a CCMA, i.e., circumventing the deep nulls problem caused by the 0th-order Bessel function.

Time-Invariant Joint Transmit and Receive Beampattern Optimization for Polarization-Subarray Based Frequency Diverse Array Radar

We propose a polarization-subarray based frequency diverse array (FDA) radar with the subarray-based FDA as the transmit (Tx) array and the polarization-sensitive subarray-based FDA (PSFDA) as the receive (Rx) array. The subarray-based FDA has the capability to achieve a single maximum beampattern at the target location, while the PSFDA can provide an extra degree of freedom to further suppress the interference and, thus, to improve the radar's signal-to-interference-plus-noise ratio (SINR). The time-dependent frequency offsets are designed for the proposed radar to realize the time-invariant beampattern at the desired target location over the whole pulse duration. To further improve the target detection performance, the time-invariant joint Tx–Rx beampattern design is considered based on the output SINR maximization. To effectively solve the nonconvex output SINR maximization problem, a suboptimal alternating optimization algorithm is proposed to iteratively optimize the FDA Tx beamforming, the PSFDA spatial pointings, and the PSFDA Rx beamforming. Numerical experiments illustrate that the time-invariant and single-maximum joint Tx–Rx beampattern at the target location is achieved. Moreover, compared to the basic FDA and logarithmic frequency offset FDA as well as conventional phased array radars, the proposed polarization-subarray based FDA radar achieves a significant SINR improvement, particularly when the desired target and the interferences are spatially indistinguishable.

Constant Modulus MIMO Radar Waveform Design With Minimum Peak Sidelobe Transmit Beampattern

This paper addresses the problem of minimum peak sidelobe transmit beampattern design for colocated multiple-input multiple-output radar systems. Two new methods are proposed to achieve it via designing constant modulus (CM) waveforms directly: One is to maximize the ratio of the minimum mainlobe level to the peak sidelobe level (PSL) without specifying pattern masks, and another is to minimize PSL under the mainlobe ripple and unspecified waveform modulus constraints. The resultant optimization problems are difficult to solve due to the coupled numerator and denominator of the quadratic fractional programming problem formulation in the former and double-sided quadratic constraints and unknown waveform modulus in the latter. For the former, we decouple the numerator and denominator by equivalent transformation and introduction of boundary-type auxiliary variables in order that feasible methods are derived to tackle them independently and efficiently. For the latter, we simplify the coupled double-sided quadratic constraints on the same waveform vector into those on different single auxiliary variables to tackle the corresponding nonconvex optimization problem. Numerical examples show that the proposed algorithms can attain sufficient low peak sidelobe beampattern levels under the CM constraints.

Computationally Simple MMSE (A-Optimal) Adaptive Beam-Pattern Design for MIMO Active Sensing Systems via a Linear-Gaussian Approximation

This paper presents an approximate minimum mean squared error (MMSE) adaptive beam-pattern design (ABD) method for MIMO active sensing systems. The proposed approximate MMSE ABD method leverages the physics of the MIMO arrays to provide a linear-Gaussian approximation that is specific to MIMO active sensing systems, and yields a computationally simple optimization problem. Computational complexity analysis confirms this theoretical reduction in the number of floating-point operations required, most notably that evaluation of the proposed approximate optimization cost function grows polynomially with the number of targets being tracked, whereas for evaluation of the exact cost the growth is exponential. Additionally, numerical results indicate that, even for a simple scenario with a single target being tracked, the proposed approximate MMSE ABD method does indeed reduce the mean squared error of target parameter estimation compared to the nonadaptive case, with a reduction in computation time of four orders of magnitude compared to exact MMSE ABD.

Separable transmit beampattern design for MIMO radars with planar colocated antennas

Multiple-input multiple-output (MIMO) radar transmit beampattern design for one-dimensional arrays has been widely studied in literatures. In this paper, transmit beampattern design is considered for two-dimensional (2D) arrays. As the size of the array is increased, the computational complexity and time for pattern optimization are increased drastically. To overcome these problems, we introduced the conditions upon which the 2D beampattern design for uniform rectangular arrays (URA) can be achieved via the product of two perpendicular transmit beampatterns of uniform linear arrays (ULA). The transmit beampattern design is accomplished under special characteristics such as minimum integrated sidelobe level or special 3 dB beamwidth in azimuth and elevation with much lower computation time.

Nested Array Sensor With Grating Lobe Suppression and Arbitrary Transmit–Receive Beampattern Synthesis

Sparse arrays have potential advantages over their regularly spaced counterparts, but they will generate undesired grating lobes. This paper investigates a sparse linear phased-array for grating lobe suppression and arbitrary transmit–receive beampattern synthesis. To achieve these aims, a modified two-level nested-array is developed, which provides increased degrees-of-freedom than a standard uniform linear array with the same number of elements via the difference co-array processing algorithm in the receiver. A least squares and iterative algorithm is also devised for linear array arbitrary transmit–receive beampattern design with minimum number of antenna elements. Moreover, the nested-array design is exploited for minimum variance distortionless response adaptive beamforming and direction finding. The effectiveness of the proposed approach is verified through extensive numerical results.

Toward Dual-functional Radar-Communication Systems: Optimal Waveform Design

We focus on a dual-functional multi-input-multi-output (MIMO) radar-communication (RadCom) system, where a single transmitter with multiple antennas communicates with downlink cellular users and detects radar targets simultaneously. Several design criteria are considered for minimizing the downlink multiuser interference. First, we consider both omnidirectional and directional beampattern design problems, where the closed-form globally optimal solutions are obtained. Based on the derived waveforms, we further consider weighted optimizations targeting a flexible tradeoff between radar and communications performance and introduce low-complexity algorithms. Moreover, to address the more practical constant modulus waveform design problem, we propose a branch-and-bound algorithm that obtains a globally optimal solution, and derive its worst-case complexity as function of the maximum iteration number. Finally, we assess the effectiveness of the proposed waveform design approaches via numerical results.

On the Statistical Resolution Limit (SRL) for Time-Reversal based MIMO radar

In the single-input multiple-output radar, the system transmits scaled (coherent) versions of a single waveform. The multiple-input multiple-output (MIMO) radar uses multiple antennas to simultaneously transmit several non-coherent waveforms and exploits multiple antennas to receive the reflected signals (echoes). This diversity in term of waveform coding allows to transmit orthogonal waveforms which enables the MIMO radar superiority in several fundamental aspects, including: improved parameter identifiability and estimation and much enhanced flexibility for transmit beam-pattern design. The context of this work is the co-located MIMO radar where the transmit and the receive arrays are close in space. In this paper, we provide a theoretical performance analysis to compare two configurations of MIMO radar: Conventional configuration and Time Reversal (TR) configuration in term of Statistical Resolution Limit (SRL). This study provides new insights on the performance gain of the TR scheme which is discussed and illustrated by appropriate simulation results depending on the receive noise level.

Constrained transmit beampattern design for colocated MIMO radar

This paper considers the constrained waveform design for Multiple-Input Multiple-Output (MIMO) radar to synthesize a desired beampattern. Specifically, resorting to SemiDefinite Programming (SDP) related technique, we first minimize Integration Sidelobe Level (ISL) to optimize the waveform covariance matrix enforcing a uniform elemental power restriction as well as a 3dB bandwidth constraint. Then, based on Least Square (LS) approach, we present the existing Cyclic Algorithm (CA) and a new Sequential Iterative Algorithm (SIA) to devise the waveform under a constant modulus constraint, and a similarity constraint to allow the designed waveform sharing the similarity feature with a given reference waveform. In particular, the proposed SIA directly optimizes the objective function and its each iteration turns the multidimensional optimization problem into multiple one-dimensional optimization problems with closed-form solutions. Finally, we assess the effectiveness of the proposed technique through numerical simulations in comparison with CA.

Constant Modulus Waveform Design for MIMO Radar Transmit Beampattern

A multiple-input multiple-output radar has great flexibility to design the transmit beampattern via selecting the probing waveform. The idea of current transmit beampattern design is to approximate the disired transimit beampattern and minimize the cross-correlation sidelobes. In this paper, under the constant modulus constraint, two algorithms are proposed to design the probing waveform directly. In the first algorithm, the optimization criterion is minimizing the squared-error between the designed beampattern and the given beampattern. Since the objective function is a nonconvex fourth-order polynomial and the constant modulus constraint can be regarded as many nonconvex quadratic equality constraints, an efficient alternating direction method of multipliers (ADMM) algorithm, whose convergence speed is very fast, is proposed to solve it. In the second algorithm, the criterion is minimizing the absolute-error between the designed beampattern and the given beampattern. This nonconvex problem can be formulated as $l_1$ -norm problem, which can be solved through a double-ADMM algorithm. Finally, we assess the performance of the two proposed algorithms via numerical results.

Location‐aided channel tracking and downlink transmission for HST massive MIMO systems

In massive multiple-input multiple-output (MIMO) high-speed train (HST) wireless communication systems (WCS), due to the large-dimension and rapidly time-varying property, channel tracking rather than conventional channel estimation is more feasible. Due to the non-linear channel variation, the channel is first decomposed into sub-channels with different angle of departure. By exploiting the property of near line of sight (LoS) and location information, Kalman filtering is used to track the channel by predicting and modifying the channel gain of the LoS sub-channel. To further improve the tracking performance, a location-based sequentially optimal beam pattern design is proposed. Since placing pilot beams for many receive antennas with different locations in time or frequency division manner costs much system resource, the authors further propose a low-complexity grouping algorithm based on location. Finally, two transmission designs based on the proposed channel tracking scheme and the location only, respectively, are introduced. Simulations validate the efficiency of the authors' proposed tracking scheme and the corresponding transmission design, which shows the importance of location information and channel tracking in massive MIMO HST-WCS.

Beam pattern design of circular antenna array via efficient biogeography-based optimization

In this paper, an efficient biogeography-based optimization (EBBO) method is proposed to synthesize the circular antenna arrays (CAA) for specific radiation beam pattern properties and null controls. The proposed method achieves the desired beam patterns by jointly optimizing the excitation currents as well as the spaces between the array elements. Three improved components, including chaotic search theory, model learning method and a new random perturbation operator, are introduced into the standard biogeography-based optimization (BBO) to improve the performance of the algorithm. Simulation results show that the maximum sidelobe level obtained by EBBO can be suppressed effectively compared with other algorithms. Moreover, the null controlling performance of EBBO is the best among the algorithms.

Fast Unit-Modulus Least Squares With Applications in Beamforming

Unit-modulus least squares (ULS) problems arise in many applications, including phase-only beamforming, sensor network localization, synchronization, phase retrieval, and radar code design. ULS formulations can always be recast as unit-modulus quadratic programs, to which semidefinite relaxation (SDR) can be applied, and is often the state-of-the-art approach. However, SDR lifts the problem dimension (i.e., the number of variables) from $N$ to $N^2$ , which drastically increases the memory burden and computational cost when the problem size is already large—e.g., when designing phase-only beamformer weights for massive multiple-input-multiple-output systems. This paper focuses on scalable first-order algorithms for the ULS problem and some of its variants. It advocates using simple gradient projection (GP) as a starting point for solving the ULS problem, establishes global convergence of GP to a Karush–Kuhn–Tucker point for this NP-hard problem, and bounds its iteration complexity. Then it proposes ULS extensions tailored to reflect practical beamformer design objectives, bringing in and exploiting new degrees of freedom to improve the beampattern designs. Simple variants of GP are proposed to deal with these extended ULS problems. Simulations are used to showcase the effectiveness of the proposed algorithms in both the plain ULS problem and in the context of phase-only beamforming.

Tractable Transmit MIMO Beampattern Design Under a Constant Modulus Constraint

Multiple-input multiple-output (MIMO) radar systems allow each antenna element to transmit a different waveform. This waveform diversity can be exploited to enhance the beampattern design, in particular, effective management of radar radiation power in directions of interest. We address the problem of designing a beampattern for MIMO radar, which in turn is determined by the transmit waveform. While unconstrained design is straightforward, a key open challenge is enforcing the constant modulus constraint on the radar waveform. It is well known that the problem of minimizing deviation of the designed beampattern from an idealized one subject to the constant modulus constraint constitutes a hard nonconvex problem. Existing methods that address constant modulus invariably lead to a stiff tradeoff between analytical tractability (achieved by relaxations and approximations) and realistic design that exactly achieves constant modulus but is computationally burdensome. A new approach is proposed in our paper, which involves solving a sequence of convex equality constrained quadratic programs, each of which has a closed form solution and such that constant modulus is achieved at convergence. We further prove that the converged solution satisfies the Karush–Kuhn–Tucker optimality conditions of the aforementioned hard nonconvex problem. We evaluate the proposed successive closed forms (SCF) algorithm against the state-of-the art MIMO beampattern design techniques in both narrowband and wideband setups and show that the SCF breaks the tradeoff between desirable performance and the associated computation cost.

On the Design of Frequency-Invariant Beampatterns With Uniform Circular Microphone Arrays

This paper deals with two critical issues about uniform circular arrays (UCAs): frequency-invariant response and steering flexibility. It focuses on some optimal design of frequency-invariant beampatterns in any desired direction along the sensor plane. The major contributions are as follows. 1) We explain how to include the steering information in the desired directivity pattern. 2) We show that the optimal approximation of the beamformer's beampattern with a UCA from a least-squares error perspective is the Jacobi–Anger expansion. 3) We develop an approach to the design of any desired symmetric directivity pattern, where the deduced beampattern is almost frequency invariant and its main beam can be pointed to any wanted direction in the sensor plane. 4) With the proposed approach, we derive an explicit form of the white noise gain (WNG) and the directivity factor (DF), and explain clearly the white noise amplification problem at low frequencies and the DF degradation at high frequencies. The analysis also indicates that increasing the number of microphones can always improve the WNG. We show that the proposed method is a generalization of circular differential microphone arrays. The relationship between the proposed method and the so-called circular harmonics beamformers is also discussed.