Frequency-invariant Beam Patterns Research Articles

Design of Differential Loudspeaker Line Array for Steerable Frequency-Invariant Beamforming

Differential beamforming has attracted much research since it can utilize an array with a small aperture size to form frequency-invariant beampatterns and achieve high directional gains. It has recently been applied to the loudspeaker line array to produce a broadside frequency-invariant radiation pattern. However, designing steerable frequency-invariant beampatterns for the loudspeaker line array has yet to be explored. This paper proposes a method to design a steerable differential beamformer with a loudspeaker line array. We first determine the target differential beampatterns according to the desired direction, the main lobe width, and the beampattern order. Then, we transform the target beampattern into the modal domain for representation. The Jacobi-Anger expansion is subsequently used to design the beamformer so that the resulting beampattern matches the target differential beampattern. Furthermore, based on the criterion of minimizing the mean square error between the synthesized beampattern and the ideal one, a multi-constraint optimization problem, which compromises between the robustness and the mean square error, is formulated to calculate the optimal desired weighting vector. Simulations and experimental results show that the proposed method can achieve steerable frequency-invariant beamforming from 300 Hz–4 kHz.

Design of fully steerable broadband beamformers with concentric circular superarrays.

Concentric circular microphone arrays have been used in a wide range of applications, such as teleconferencing systems and smarthome devices for speech signal acquisition. Such arrays are generally designed with omnidirectional sensors, and the associated beamformers are fully steerable but only in the sensors' plane. If operated in the three-dimensional space, the performance of those arrays would suffer from significant degradation if the sound sources are out of the sensors' plane, which happens due to the incomplete spatial sampling of the sound field. This paper addresses this issue by presenting a new method to design concentric circular microphone arrays using both omnidirectional microphones and bidirectional microphones (directional sensors with dipole-shaped patterns). Such arrays are referred to as superarrays as they are able to achieve higher array gain as compared to their traditional counterparts with omnidirectional sensors. It is shown that, with the use of bidirectional microphones, the spatial harmonic components that are missing in the traditional arrays are compensated back. A beamforming method is then presented to design beamformers that can achieve frequency-invariant beampatterns with high directivity and are fully steerable in the three-dimensional space. Simulations and real experiments validate the effectiveness and good properties of the presented method.

Eigenbeam-space transformation based steerable differential beamforming for linear arrays

Since linear differential arrays with small apertures can obtain high directivity and frequency-independent spatial response, much attention has been paid to the design of differential beamformers steered to the endfire direction. Although some works focus on the design of steerable differential beamformers with linear arrays, the synthesis of steerable frequency-invariant beampatterns by linear arrays still needs to be investigated. In this paper, an eigenbeam-space transformation based steerable differential beamforming is proposed for linear arrays. The major contributions are as follows. (1) For the beamformer steered away from the endfire direction, an interference suppression constraint might be imposed to ensure a valid beampattern in the situation where the interference from the endfire direction is very strong, which is not considered in the state-of-the-art approaches of the literature. (2) The synthesis of steerable differential beamformers is formulated as an optimization problem in the eigenbeam-space domain and the optimization problem is solved in a closed form. By experimentally determining the best pair of integers mt and N′, the proposed hybrid least-squares-error solution obtained in the eigenbeam-space domain has shown some advantages over the null constrained method and the weighted least-squares method implemented in the element-space domain.

Broadband Beamforming Using the Least Square Method Improved via Adaptive Diagonal Loading

The least square method (LSM) is a commonly used beamforming method. However, it limits the beamforming frequency bandwidth and is vulnerable to noise; therefore, diagonal loading is used to overcome these limitations. The diagonal loading proposed in this study can be used for broadband beamforming in cases in which noise does not exist, and it makes the loading factor adaptive to a singular value distribution. The simulation results show that the proposed method works stably above the beamforming frequency limit, even for an extended beamforming frequency. We expect that the proposed method can be used to form frequency-invariant beam patterns used in monopulse algorithms to detect signals of unknown frequencies.

An optimization method for frequency-invariant beamforming with arbitrary sensor arrays

This paper proposes a beampattern optimization method for frequency-invariant beamforming with arbitrary sensor arrays, which is an extension of the previously proposed method for circular sensor arrays. Based on the criterion of minimizing the mean square error between the synthesized beampatterns at each frequency and the desired beampattern, the relationship between the weighting vectors and the desired weighting vector can be accurately obtained. Then, the relevant performance parameters, such as the directivity factor, error sensitivity function and minimum mean square error, can all be expressed as the accurate closed-form functions of the desired weighting vector. With the above conclusions, a multi-constraint optimization problem is formulated to calculate the optimal desired weighting vector under different constraints. The optimal frequency-invariant beampatterns and their performance parameters can be readily achieved using the derived functions. Thus, the proposed method can not only obtain the optimal desired weighting vector, but also achieve the broadband frequency-invariant beampatterns with only one optimization calculation of the desired weighting vector. Simulations and experimental results show that the proposed method can provide a good trade-off among the directivity, robustness, and frequency invariance.

Synthesis of Wideband Frequency-Invariant Beam Patterns for Nonuniformly Spaced Arrays by Generalized Alternating Projection Approach

This work generalizes the alternating projection approach (APA) to synthesize wideband frequency-invariant (FI) beam patterns for linear nonuniformly spaced arrays (NUSAs). Different from the conventional APA where excitation weights are synthesized for a desired pattern at single frequency, the generalized APA intends to find appropriate finite-impulse response (FIR) filter coefficients for each antenna element to provide the required wideband frequency-dependent excitation for a desired wideband FI pattern, and two modification techniques called mainlobe FI modification and wideband sidelobe control are introduced in each iteration to successively update the FI pattern performance. Several examples for synthesizing wideband FI beam patterns for NUSAs in different applications are conducted to verify the effectiveness and advantages of the proposed generalized APA.

Design of 2D and 3D Differential Microphone Arrays With a Multistage Framework

Differential microphone arrays (DMAs) have demonstrated a great potential for high-fidelity acoustic and speech signal acquisition in a wide range of applications since such arrays are able to achieve frequency-invariant beampatterns with high directivity. Consequently, a great number of efforts have been devoted to the design of DMAs and the associated beamformers in the literature. However, most of the methods only work for arrays with particular topologies, e.g., linear, circular, concentric circular, and spherical ones. How to design general two-dimensional (2D) and three-dimensional (3D) DMAs that can measure the desired differential sound field and form the desired spatial response in the 3D space remains an unsolved problem. This paper investigates this problem and presents a multistage design approach. The major contributions of this work are as follows. First, we reexamine the differentials of the acoustic pressure field in the 3D space and derive the general expression of the directivity patterns resulting from the spatial differential operation, which serves as the foundation for differential beamforming with 2D or 3D microphone arrays. Second, we present a multistage approach to the design of 2D and 3D DMAs, and deduce the relationship between the global beamformer and the beamformers at different stages as well as the relationship between their beampatterns. Third, several algorithms are presented for the design of differential as well as robust differential beamformers in this multistage framework. Simulation results validate the proposed approach and justifies its properties.

On the Design of 3D Steerable Beamformers With Uniform Concentric Circular Microphone Arrays

Circular microphone arrays (CMAs) and concentric CMAs (CCMAs) have been used in a wide range of applications such as smartspeakers and teleconferencing systems because of their flexible steering ability. Although many efforts have been devoted to beamforming with CCMAs, most existing methods consider only the 2-dimensional (2D) case and assume that the sound sources of interest are in the same plane as the sensor array (generally the horizontal plane), which often does not hold true in practical applications. This paper deals with the problem of beamforming with uniform CCMAs (UCCMAs) in the 3-dimensional (3D) space to control the steering of the spatial response and meanwhile form frequency-invariant beampatterns for processing broadband acoustic and speech signals. The major contributions of this work are summarized as follows: 1) it presents an analysis based on the spherical harmonics decomposition about the $N$ th-order optimal and steerable directivity patterns; 2) a beamforming method is developed in which the beamformer's coefficients are identified by solving a linear system of equations formed by approximating the $N$ th-order optimal target beampattern with the beamformer's beampattern while the resulting beampattern can be steered flexibly in the 3D space; 3) the sufficient and necessary condition on the array geometry and sensors’ placement are given to ensure that the beamformer exists and is unique; and 4) the analytical forms of the directivity factor (DF) and white noise gain (WNG) of the resulting beamformer is given and discussion is presented on what conditions irregularities (deep nulls) in WNG and DF may occur. Simulations are provided to illustrate the property of the developed beamforming methods.

On a Particular Family of Differential Beamformers With Cardioid-Like and No-Null Patterns

Differential microphone arrays (DMAs), which are responsive to the differential acoustic pressure fields, have been used in a wide range of applications related to audio and speech. The core part of a DMA is the so-called differential beamformer, which is generally designed by placing a number of nulls in its beampattern to attenuate noise from some directions. But the presence of these nulls may cause some great issues, e.g., leading to suboptimal performance if the interference/noise is incident from directions other than the nulls' directions, and making the beamformer less robust to sensors' self noise and array imperfections. To overcome these problems, this letter is devoted to the design of differential beamformers with no nulls in its beampattern. A design method and its multistage implementation are presented and analyzed. An improved solution is then developed, which is able to form frequency-invariant beampatterns with no nulls in the frequency range of speech signals. Simulations are provided to illustrate the properties of the developed methods.

Differential Beamforming From the Beampattern Factorization Perspective

Differential beamformers have demonstrated a great potential in forming frequency-invariant beampatterns and achieving high directivity factors. Most conventional approaches design differential beamformers in such a way that their beampatterns resemble a desired or target beampattern. In this paper, we show how to design differential beamformers by simply taking advantage of the fact that the beampattern is actually a particular form of an exponential polynomial. Thanks to this quite obvious formulation, a target beampattern is not really needed while the zeros of the exponential polynomial and/or its factorization are fully exploited. The advantage of this factorization is twofold. First, it gives the relation between the beamformer and the roots of the polynomial, so the former can be directly determined from the latter, which seems natural and convenient. Second, based on this factorization, we propose a new formulation of the beamforming filter, which decomposes the filter into shorter ones with the Kronecker product. This formulation is very general, and many well-known beamformers such as the differential and delay-and-sum (DS) ones can be derived from it. Furthermore, the new formulation allows one to combine different kinds of beamformers together, which gives a great flexibility in forming different beampatterns and achieving a compromise among the directivity factor (DF), white noise gain (WNG), and frequency invariance.

Superdirective Robust Algorithms’ Comparison for Linear Arrays

Frequency-invariant beam patterns are often required by systems using an array of sensors to process broadband signals. In some experimental conditions (small devices for underwater acoustic communication), the array spatial aperture is shorter than the involved wavelengths. In these conditions, superdirective beamforming is essential for an efficient system. We present a comparison between two methods that deal with a data-independent beamformer based on a filter-and-sum structure. Both methods (the first one numerical, the second one analytic) formulate a mathematical convex minimization problem, in which the variables to be optimized are the filters coefficients or frequency responses. The goal of the optimization is to obtain a frequency invariant superdirective beamforming with a tunable tradeoff between directivity and frequency-invariance. We compare pros and cons of both methods measured through quantitative metrics to wrap up conclusions and further proposed investigations.

Design of an Efficient Wideband Beamformer using OPSO

Wideband beamforming is a technique which is used to produce frequency invariant beampatterns over a wide signal bandwidth. In this paper, orthogonal particle swarm optimization (OPSO) based wideband beamformer is designed to achieve wideband beamforming. Simulations are carried out for two different scenarios to illustrate the capability of the algorithm for wideband beamforming. The beampatterns obtained from three different methods that are standard Frost algorithm, particle swarm optimization (PSO) and OPSO are compared by various parameters such as level of interference, deviation in interference angle and peak side lobe level. It is shown that by using OPSO, SLL (Sidelobe Level), null depth as well as angle deviation in nulls direction can be reduced as compared to PSO and Frost algorithms.

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.

Reducing the Number of Elements in the Synthesis of a Broadband Linear Array With Multiple Simultaneous Frequency-Invariant Beam Patterns

The problem of reducing the number of elements in a broadband linear array with multiple simultaneous crossover frequency-invariant (FI) patterns is considered. Different from the single FI pattern array case, every element channel in the multiple FI pattern array is divided and followed by multiple finite-impulse-response (FIR) filters, and each of the multiple FIR filters has a set of coefficients. In this situation, a collective filter coefficient vector and its energy bound are introduced for each element, and then the problem of reducing the number of elements is transformed as minimizing the number of active collective filter coefficient vectors. In addition, the radiation characteristics including beam pointing direction, mainlobe FI property, sidelobe level, and space-frequency notching requirement for each of the multiple patterns can be formulated as multiple convex constraints. The whole synthesis method is implemented by performing an iterative second-order cone programming (SOCP). This method can be considered as a significant extension of the original SOCP for synthesizing broadband sparse array with single FI pattern. Numerical synthesis results show that the proposed method by synthesizing multiple discretized crossover FI patterns can save more elements than the original iterative SOCP by using a single continuously scannable FI pattern for covering the same space range. Moreover, even for multiple FI-patterns case with complicated space-frequency notching, the proposed method is still effective in the reduction of the number of elements.

Beamforming based on null-steering with small spacing linear microphone arrays.

This paper develops an approach to beamforming with small spacing uniform linear microphone arrays based on the null-steering (NS) principle. It first formulates the beamforming problem from the conventional mean-squared error (MSE) criterion and its normalized version. Several NS algorithms are then derived for beamforming with the constraint of placing nulls to either a single direction or multiple angles. The difference and relationships between different algorithms are discussed and their performances are evaluated. These algorithms can be used to design either fixed or adaptive beamformers. In the former case, the resulting beamformers work as differential microphone arrays (DMAs) since they exhibit frequency-invariant beampatterns and their directivity factors are high with a given number of sensors. In the latter case, the resulting beamformers can be viewed as a combination of DMAs and single-channel noise reduction since they do not only exhibit frequency-invariant beampatterns but also can achieve noise reduction based on the noise statistics.

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.

Reduced-Order Robust Superdirective Beamforming With Uniform Linear Microphone Arrays

Sensor arrays for audio and speech signal acquisition are generally required to have frequency-invariant beampatterns to avoid adding spectral distortion to the broadband signals of interest. One way to obtain frequency-invariant beampatterns is via superdirective beamforming. However, traditional superdirective beamformers may cause significant white noise amplification (particularly at low frequencies), making them sensitive to uncorrelated white noise. To circumvent the problem of white noise amplification, a method was developed to find the superdirective beamforming filter with a constraint on the white noise gain (WNG), leading to the so-called WNG-constrained superdirective beamformer. But this method damages the frequency invariance of the beampattern. In this paper, we develop a flatness-constrained robust superdirective beamformer. We divide the overall beamformer into two subbeamformers, which are convolved together: one subbeamformer forms a lower order superdirective beampattern while the other attempts to improve the WNG. We show that this robust approach can improve the WNG while limiting the frequency dependency of the beampattern at the same time.

Differential beamforming with circular microphone arrays

Differential beamforming cannot only achieve high directional gain but also can form frequency-invariant beampatterns. Therefore, it has the great potential to be widely used in voice communications to process broadband speech signals. The performance of a differential beamformer in terms of directivity factor (DF), white noise gain (WNG), and frequency invariance of the beampattern depends on many factors including the array geometry, the number of sensors, the inter-sensor spacing, etc. In the literature, most studies on differential beamforming have been focused on linear microphone arrays and not much efforts have been devoted to other array geometries. This paper is devoted to the study of circular microphone arrays. In comparison with linear arrays whose spatial response is a function of the steering angle, circular microphone arrays can have the same spatial response along many different directions. The focal point of this paper is on differential beamforming with uniform circular microphone arrays. The main topics addressed include: (1) major properties of circular microphone arrays, (2) how to design differential beamformers with different beampatterns, (3) how to design differential beamformers with different orders, and (4) how to achieve a good control of the white noise amplification problem, high directional gains, and frequency-independent responses.

Theoretical Analysis of Differential Microphone Array Beamforming and an Improved Solution

Differential microphone arrays (DMAs), which are responsive to the differential sound pressure field, have attracted much attention due to their properties of frequency-invariant beampatterns, small apertures, and potential of maximum directivity. Traditionally, DMAs are designed and implemented in a multistage (cascade) way, where a proper time delay is used in each stage to form a beampattern of interest. Recently, it was reported that DMAs can be designed by solving a linear system of equations formed from the information about the nulls of the desired beampattern. This paper deals with the problem of beamforming with linear DMAs. Its major contributions are as follows. 1) By using the spatial ${\cal Z}$ transform, we present some theoretical analysis of both the traditional cascade and new null-constrained DMA beamforming. It is shown that the cascade and null-constrained DMAs of the same order with the same number of sensors are theoretically identical. 2) We develop a two-stage approach to the study of the robust DMA beamformer, which is based on the principle of maximizing the white noise gain (WNG). The first-stage of this approach is in the structure of the traditional non-robust DMA while the second-stage filter is optimized for improving the WNG. 3) Using the two-stage approach, we show that the robust DMA beamformer may introduce extra nulls in the beampattern at high frequencies; particularly, it introduces $M - N - 1$ extra nulls if the interelement spacing is equal to half of the wavelength, where $M$ and $N$ are the number of sensors and the DMA order, respectively. 4) We develop a method that can solve the extra-null problem while maximizing the WNG in robust DMA beamforming, i.e., a robust solution with a frequency-invariant beampattern.

Design of robust differential microphone arrays with orthogonal polynomials.

Differential microphone arrays have the potential to be widely deployed in hands-free communication systems thanks to their frequency-invariant beampatterns, high directivity factors, and small apertures. Traditionally, they are designed and implemented in a multistage way with uniform linear geometries. This paper presents an approach to the design of differential microphone arrays with orthogonal polynomials, more specifically with Jacobi polynomials. It first shows how to express the beampatterns as a function of orthogonal polynomials. Then several differential beamformers are derived and their performance depends on the parameters of the Jacobi polynomials. Simulations show the great flexibility of the proposed method in terms of designing any order differential microphone arrays with different beampatterns and controlling white noise gain.