Frequency Invariant Beam Pattern Research Articles

Design of Robust Broadband Frequency-Invariant Broadside Beampatterns for the Differential Loudspeaker Array

The directional loudspeaker array has various applications due to its capability to direct sound generation towards the target listener and reduce noise pollution. Differential beamforming has recently been applied to the loudspeaker line array to produce a broadside frequency-invariant radiation pattern. However, the existing methods cannot achieve a compromise between robustness and broadband frequency-invariant beampattern preservation. This paper proposed a robust broadband differential beamforming design to allow the loudspeaker line array to radiate broadside frequency-invariant radiation patterns with robustness. Specifically, we propose a method to determine the ideal broadside differential beampattern by combining multiple criteria, namely null positions, maximizing the directivity factor, and achieving a desired beampattern with equal sidelobes. We derive the above ideal broadside differential beampattern into the target beampattern in the modal domain. We propose a robust modal matching method with Tikhonov regularization to optimize the loudspeaker weights in the modal domain. Simulations and experiments show improved frequency-invariant broadside beamforming over the 250–4k Hz frequency range compared with the existing modal matching and null-constrained methods.

Low-complexity frequency-invariant beampattern synthesis using accurate response control for speech extraction

Beampattern synthesis generally designs weight parameters to form a main beam towards the direction of interest while suppressing interference and environmental noise simultaneously, playing a critical role in fixed beamforming techniques. In practical applications, when the position of the interference source changes, the filter design requirements will correspondingly change, thus necessitating accurate response control and rapid update capabilities. A low-complexity method is also highly desired in broadband beampattern synthesis, especially for time-domain beamformers with a relatively large number of weight parameters. To achieve rapid update and reduce the complexity for time-domain beamformers, this paper proposes a low-complexity frequency-invariant beampattern synthesis method using accurate response control. By introducing the null-forming scheme of adaptive beamformers, a virtual interference-plus-noise covariance matrix is constructed to precisely control the beampattern. Additionally, the spatial response variation is considered in the optimization problem for a constant mainlobe pattern, avoiding the complexity of determining a desired beampattern. Compared with the existing algorithms based on the interior-point method and the alternating direction method of multipliers, the proposed method reduces the computational complexity of each iteration, showing higher efficiency in broadband beampattern synthesis. Furthermore, the proposed method adjusts the beamformer flexibly on the basis of a pre-designed beampattern, achieving fast beampattern update when the acoustic environment changes. Numerical simulations on beam patterns and experimental results on speech extraction demonstrate better interference suppression performance of the beamformers designed by the proposed method.

Frequency-Invariant Beampattern Synthesis for Wideband MIMO Radar Under Nulling and PAPR Control

Frequency-Invariant Beampattern Synthesis for Wideband MIMO Radar Under Nulling and PAPR Control

Mode matching-based beamforming with frequency-wise truncation order for concentric circular differential microphone arrays.

It is desired that a fixed beamformer should maintain the frequency-invariant beampattern and achieve the high white noise gain (WNG), i.e., high robustness against the mismatch in practice. However, existing methods for the design of concentric circular differential microphone arrays (CCDMAs) cannot achieve a compromise between the high robustness and the frequency-invariant beampattern. To address this problem, a new analytical expression for the synthesized beampattern of CCDMAs is derived without any truncation error. Then CCDMAs are designed by matching mode coefficients of the approximated synthesized beampattern to that of the target differential beampattern, where an adjustable truncation order is utilized to enable a trade-off between the robustness and the beampattern distortion. A simple and effective procedure is presented to determine the frequency-wise truncation order. The proposed method reduces to three existing methods, i.e., the Jacobi-Anger method, the improved Jacobi-Anger method, and the minimum mean square error-based method, for a fixed truncation order, which in turn establishes a close connection with these methods and provides a unified view on the design of the CCDMAs. The superiority of the proposed method in terms of robustness and beampattern distortion is demonstrated through computer simulations.

Accurate Spatial-Frequency Response Controllable Frequency Invariant Beampattern Synthesis With Minimum Beamwidth

This communication proposes a new relaxation variable based cost function for the problem of frequency invariant (FI) beampattern synthesis. Two constraints in the spatial and frequency domains are imposed on the optimization problem. The objective of sparsifying the relaxation variable is exploited in constructing the constraint in the spatial domain, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -spatial response variation (SRV) is introduced in the frequency domain constraint to reduce frequency variance. These constraints enable a desirable trade-off between beamwidth and FI properties while achieving accurate spatial-frequency response control. Since the resulting optimization problem is sparse and non-convex, it is then proposed to be solved by exploiting the reweighted- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm in this paper. The remarked feature of the new approach is the improved target detection and anti-interference performance in radar and communication systems without waveform distortion. Simulation results validate the effectiveness of the proposed method.

Design of Planar Differential Microphone Array Beampatterns with Controllable Mainlobe Beamwidth and Sidelobe Level.

The differential microphone array, or differential beamformer, has attracted much attention for its frequency-invariant beampattern, high directivity factor and compact size. In this work, the design of differential beamformers with small inter-element spacing planar microphone arrays is concerned. In order to exactly control the main lobe beamwidth and sidelobe level and obtain minimum main lobe beamwidth with a given sidelobe level, we design the desired beampattern by applying the Chebyshev polynomials at first, via exploiting the structure of the frequency-independent beampattern of a theoretical Nth-order differential beamformer. Next, the so-called null constrained and least square beamformers, which can obtain approximately frequency-invariant beampattern at relatively low frequencies and can be steered to any direction without beampattern distortion, are proposed based on planar microphone arrays to approximate the designed desired beampatterns. Then, for dealing with the white noise amplification at low-frequency bands and beampattern divergence problems at high-frequency bands of the null constrained and least square beamformers, the so-called minimum norm and combined solutions are deduced, which can compromise among the white noise gain, directivity factor and beampattern distortion flexibly. Preliminary simulation results illustrate the properties and advantages of the proposed differential beamformers.

Frequency-invariant beamformer design via ADPM approach

Frequency-invariant (FI) beamformers play an important role in suppressing speech waveform distortions. Due to the perfect FI beampattern (FIB), differential microphone arrays (DMAs) have been widely used in practical applications like voice communication and human-machine interface systems. Superdirective beamforming generated by DMAs have many useful properties but suffer white noise amplification. To address this drawback, we formulate a least squares broadband FI problem, under the white noise output power constraints, to improve robustness of superdirective beamformers. The problem is challenging to solve since FI beamformers are designed on broadband and initialization parameters corresponding to each frequency are different. We devise broadband beampattern synthesis algorithm based on alternating direction penalty method (ADPM), which utilizes the relationship between residuals and penalty terms to reduce the iteration number under the improved framework of alternating direction method of multipliers (ADMM). The proposed ADPM method can decompose the optimization problem into multi-block convex optimization problems and solve them separately. The fast convergence property of our solution is demonstrated via numerical simulations.

Differential Error Feedback Active Noise Control With the Auxiliary Filter Based Mapping Method

This letter proposes a differential error feedback active noise control (FBANC) system in an open end duct to mitigate interferences from its downstream. The interference waves entering the duct cause disturbance to the conventional FBANC controller and even result in divergence of the control filter. This is due to the omni-directivity of the error microphone. The differential microphone array (DMA) can be constructed in a compact size using only one more omni-directional microphone. The DMA forms a frequency-invariant beampattern that presents configurable nulls. When the output of the DMA is used as the error signal, the null can be designed to enhance the robustness of the FBANC system against interferences. However, this differential error signal is converted from the sound pressure gradient instead of the sound pressure. The DMA’s location does not precisely indicate the control point where optimum noise reduction has been achieved. To solve this problem, an auxiliary filter based mapping (AFMap) method is developed to map the differential error signal to the location of an omni-directional microphone in the DMA. Experiment results demonstrate that the proposed differential error FBANC system is much less sensitive to interferences than the conventional FBANC system, and the AFMap method can ensure optimum noise reduction occurring at the target control point.

Frequency-Invariant Beam Pattern Nulling Based on Weighted Pattern Summation

Frequency-Invariant Beam Pattern Nulling Based on Weighted Pattern Summation

Efficient Synthesis of Filter-and-Sum Array With Scanned Wideband Frequency-Invariant Beam Pattern and Space-Frequency Notching

This work generalizes the Fourier transform (FT)-based frequency-invariant beamforming (FIB) method to the synthesis of scanned frequency-invariant (FI) beam pattern with space-frequency notching for an array with non-isotropic elements. Wideband FI pattern characteristics are described by using multiple reference sub-band FI patterns. By applying fast Fourier transform (FFT) on the combination of these multiple reference sub-band FI patterns, a wideband excitation distribution can be generated. Based on this excitation distribution, we construct a new wideband excitation distribution that is conjugate-symmetric about zero frequency, so that real-valued finite-impulse-response (FIR) filter coefficients can be obtained by applying FFT on the constructed distribution. Two numerical examples are introduced to show the effectiveness and efficiency of the proposed method.

Wideband frequency-invariant beamforming with dynamic range ratio constraints

This paper focuses on the optimization design of weighting coefficients of Frost-structural beamformer to obtain a frequency-invariant (FI) beampattern. To ensure the normal operation of beamformer in transmitting links with limited dynamic range, the design is modeled by minimizing the weighted sum of the beampattern matching error and spatial response variation under the constraints of dynamic range ratio (DRR) and linear phase. Next, we propose a novel iterative algorithm to solve the resulting high-dimensional non-convex optimization problems via alternately handling a tractable problem and a multi-constrained convex subproblem. To speed up, we reformulate this multi-constrained subproblem into a new alternating direction method of multipliers form with a series of low-dimensional and less-constrained subproblems, and deduce the closed-form solutions to each subproblem. Numerical results exhibit the good convergence of proposed algorithm and the excellent FI of obtained beampattern under the DRR constraints.

Efficient and Accurate Frequency-Invariant Beam Pattern Synthesis Utilizing Iterative Spatiotemporal Fourier Transform

An iterative spatiotemporal Fourier transform (STFT) method is presented to efficiently design finite-impulse-response (FIR) filter coefficients for generating a desired frequency-invariant (FI) beam pattern of an antenna array. In this method, by introducing the concepts of normalized temporal angular frequency and spatial angular frequency, the broadband pattern is treated as a spatiotemporal spectral distribution. The relationship between the FIR coefficient distribution and the spatiotemporal spectral distribution can be built as an STFT, and consequently the 2-D fast Fourier transform (2D-FFT) and inverse 2D-FFT (2D-IFFT) can be utilized to efficiently accomplish the transformation between the FIR coefficient distribution and the spatiotemporal spectral distribution. Thus, the proposed synthesis method starts from an initial spatiotemporal spectral distribution and then adopt an iterative modification-and-transformation strategy to successively update the obtained spatiotemporal spectral distribution and the corresponding FIR coefficients. Two kinds of pattern modification techniques including the mainlobe FI modification and broadband sidelobe control are adopted in each iteration. Several examples for synthesizing different FI patterns are conducted. Synthesis results show that the proposed method can obtain much better FI pattern performance in terms of both mainlobe FI property and sidelobe control than the original FT method whilst costing less CPU time than the convex optimization method especially for the case of large FI arrays.

Time-domain frequency-invariant beampattern synthesis via alternating direction method of multipliers.

This paper addresses time-domain frequency-invariant beampattern synthesis under multiple constraints. Mainlobe spatial response variation is introduced to characterize the frequency invariance. Peak constraints and norm constraints are imposed to control the sidelobe response and robustness of the beamformer, respectively. Such constraints are separated with the help of the alternating direction method of multipliers (ADMM), and an iterative solution is derived. Unlike the traditional interior-point approach with cubic complexity, the proposed algorithm shows quadratic complexity in a single iteration. Simulation results demonstrate that the proposed algorithm can achieve satisfactory performance in less time.

Robust Indoor Speaker Localization in the Circular Harmonic Domain

In this article, modal signal processing using circular sensor arrays has been shown to be an attractive way for speaker localization, due to the fact that it enables high flexibility in sound field analysis, which inherently supports wideband acoustic sources and provides a frequency invariant beampattern. However, the existing circular harmonic direction-of-arrival (DOA) estimation approaches suffer from the following problems: first, depending strongly on the characteristics of sensor arrays; second, the presence of singularity when using Bessel function of the first kind; and third, being sensitive to noisy and reverberant environments. To combat these problems, this article proposes a robust indoor speaker localization method that incorporates circular harmonic pseudointensity vector through joining the least-squares decomposition and the spatial processing with a small-sized sensor array. The root mean square error of DOA estimation of the proposed method is compared with those published approaches in simulations for different noise levels and reverberation times. In addition, the superior performance of the proposed indoor speaker localization method is validated by using speech recordings in a real-world acoustic environment.

On the design of differential beamformers with arbitrary planar microphone array geometry.

This letter deals with the problem of differential beamforming with microphone arrays of arbitrary planar geometry. By approximating the beampattern with the Jacobi-Anger expansion, it develops an algorithm that can form any specified frequency-invariant beampattern with a microphone array of any planar geometry as long as the sensors' coordinates are given and the spacing between neighboring sensors is smaller than the smallest wavelength. This method is rather general and it can be used to design differential beamformers with linear, circular, and concentric circular differential microphone arrays as well as differential arrays of arbitrary planar geometry where sensors are placed in any specified positions.

Robust Frequency Invariant Beamforming with Low Sidelobe for Speech Enhancement

Frequency invariant beamformers (FIBs) are widely used in speech enhancement and source localization. There are two traditional optimization methods for FIB design. The first one is convex optimization, which is simple but the frequency invariant characteristic of the beam pattern is poor with respect to frequency band of five octaves. The least squares (LS) approach using spatial response variation (SRV) constraint is another optimization method. Although, it can provide good frequency invariant property, it usually couldn’t be used in speech enhancement for its lack of weight norm constraint which is related to the robustness of a beamformer. In this paper, a robust wideband beamforming method with a constant beamwidth is proposed. The frequency invariant beam pattern is achieved by resolving an optimization problem of the SRV constraint to cover speech frequency band. With the control of sidelobe level, it is available for the frequency invariant beamformer (FIB) to prevent distortion of interference from the undesirable direction. The approach is completed in time-domain by placing tapped delay lines(TDL) and finite impulse response (FIR) filter at the output of each sensor which is more convenient than the Frost processor. By invoking the weight norm constraint, the robustness of the beamformer is further improved against random errors. Experiment results show that the proposed method has a constant beamwidth and almost the same white noise gain as traditional delay-and-sum (DAS) beamformer.

Robust Superdirective Frequency-Invariant Beamforming for Circular Sensor Arrays

This letter proposes a robust superdirective frequency-invariant beamforming method for circular sensor arrays. First, the beampattern is accurately transformed into an equivalent form using the property of circular arrays, and the expressions of directivity factor and white noise gain are correspondingly modified. Second, a closed-form weighting vector is accurately derived based on the minimum mean square error approximation to a desired superdirective beampattern that is expressed in a general form. When the number of elements is sufficiently large, the weighting vector can be further simplified, and the white noise gain is also modified to a concise form that is found to be an increasing function of the number of elements in the low-frequency range. Simulations and experimental results show that this method can provide a robust superdirective frequency-invariant beampattern over a wide frequency band given a suitable number of elements, and it can be readily applied to process broadband signals in practice using small circular sensor arrays.

Special issue: Waveform diversity: part II

It is our great pleasure and honor to introduce the second of two special issues of the IEEE Aerospace and Electronic Systems Magazine on Waveform Diversity. The first paper,Application of Waveform Weighting for a Frequency-Invariant Transmit Beampattern,by Uysal et al. discusses a proposed method for implementing wideband beamforming in wideband active-array systems where precise beampatterns are desired. The authors use their beamforming method to design and synthesize a multi-channel set of waveforms that enables wideband beamforming while achieving a frequency-invariant beampattern with low sidelobes. In combination with digital predistortion, the weighted beamforming method has yielded effective results for LFM and P4 polyphasecoded waveforms in simulations and hardware measurements. Consequently, the authors feel that their wideband beamforming method has potential application in any wideband active array system when a precise beampattern is desired.

Design of Directivity Patterns with a Unique Null of Maximum Multiplicity

Differential beamforming is one of the most popular beamforming approaches, which has the great potential to form frequency-invariant directivity patterns. In this paper, we study the design of beampatterns with multiple nulls in the same direction, which is clearly different from the design of beampatterns with distinct nulls. Our contributions are as follows. First, we show how to constrain multiple nulls to the same direction and design the desired beampattern with both the traditional and robust approaches. Second, we derive an explicit form of the white noise gain (WNG) of the traditional approach as a function of the frequency, interelement spacing, and null direction, which shows that the cardioid is the optimal beampattern as far as the WNG is concerned. Third, we prove that the WNG improvement of the robust approach rarely depends on the null direction at low frequencies. Finally, considering the fact that the robust differential beamforming approach may produce a frequency-dependent beampattern while improving the WNG, we develop a weighted-norm approach that can make a good compromise between the robustness of differential beamforming with respect to white noise and the frequency-invariant beampattern. The performance of the developed approach is verified by simulations.

Synthesis of Sparse Arrays With Frequency-Invariant-Focused Beam Patterns Under Accurate Sidelobe Control by Iterative Second-Order Cone Programming

It is shown in this communication that the problem of synthesizing a frequency-invariant (FI) beam pattern with as few elements as possible can be transformed into a sequence of weighted $\ell_1$ optimizations under multiple convex constraints. The objective of the weighted $\ell_1$ optimization is to minimize the number of radiating elements. For the filter-and-sum beamforming structure, this is equivalent to minimizing the number of filters, each with a set of optimized coefficients. The multiple constraints are adopted to individually control the mainlobe and sidelobe pattern characteristics, so as to preserve the FI property in the mainlobe region while satisfying a given upper bound in the sidelobe region. The WNG constraint can also be incorporated into the proposed method to enhance the synthesis robustness. This method can be easily implemented by iterative second-order cone programming (SOCP), and only few iterations are required to reach the convergence. A set of examples for the synthesis of FI patterns with uniform sidelobe level (SLL) or multiple nulls, scannable FI patterns, and the synthesis of FI pattern for an arc array with directive elements, is presented to validate the effectiveness and advantages of the proposed method. The element saving is about 23.3%–50% for the test cases.