Frequency Invariant Beamforming 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.

Sparse microphone array design for frequency-invariant beamforming via group [formula omitted]-norm optimization

Frequency-invariant (FI) beamforming aims at recovering signals without any distortion and maintaining spatial selectivity over the entire bandwidth. However, most existing FI beamforming (FIB)methods consider the weighted ℓ1-norm or modified ℓ1-norm of the filters as the objective function for sparse FIB design, which cannot assure the optimal sparse solution. To deal with this drawback, an optimization problem for the FIB design of sparse microphone array in terms of ℓp-norm (0<p<1) minimization is formulated, where distortion-less response, mainlobe and sidelobe constraints are considered. Due to the existence of the ℓp-norm objective function, the resultant problem is nonconvex, and therefore, an alternating direction method of multipliers (ADMM) algorithm is devised. Specifically, the corresponding problem is decomposed into multi-block subproblems to determine all unknown variables separately. Then, the most dominant sensor positions for each frequency are determined by using the principal component analysis (PCA) and K-mean clustering algorithm. Numerical examples show that the proposed design achieves good directivity factor, frequency invariance and better sparsity.

A note on the design of frequency-invariant beamforming with uniform concentric circular microphone array

Compared to the frequency-invariant beamformer (FIB) with uniform circular microphone arrays, the FIB with uniform concentric circular microphone arrays (UCCAs) can mitigate the deep nulls problem in the white noise gain (WNG) and the directivity factor (DF). The WNG of the FIB-UCCAs can be improved by increasing the number of microphones. However, certain Jacobi-Anger expansion based methods for FIB-UCCAs utilize a linearly transformed version of the beamformer coefficients to optimize the WNG, which may result in severe beampattern distortion at high frequencies. In this paper, we present a beamformer design method that is based on the Jacobi-Anger expansion but optimizes the WNG directly. We analyze the effect of the truncation error of Jacobi-Anger expansion and demonstrate that the proposed method can effectively reduce the beampattern distortion. Computer simulations illustrate the performance advantages of the developed method in terms of the DF and WNG.

Design of frequency-invariant uniform concentric circular arrays with first-order directional microphones

Most of the existing frequency-invariant beamformers (FIBs) with circular arrays are implemented with omnidirectional microphones, where uniform circular arrays (UCAs) and uniform concentric circular arrays (UCCAs) have been widely utilized. However, the resulting FIB usually suffers from the deep nulls problem in the white noise gain (WNG) and the white noise amplification problem at low frequencies even if the order of the target beampattern is one. Due to its simple implementation, first-order directional microphones are adopted to improve the robustness of the FIB with UCCAs (FIB-UCCAs) in this paper. We exploit the circular harmonics decomposition rather than the Jacobi–Anger expansion to approximate the beampattern of the FIB-UCCAs with first-order directional microphones, because the directivity pattern of directional microphones is related to the direction of the signal. We show that the FIB-UCCAs can completely avoid the deep nulls problem in the WNG as long as the ring with the largest radius utilizes first-order directional microphones. In addition, the white noise amplification problem at low frequencies can be greatly alleviated if first-order directional microphones are utilized. Specifically, we show that the inherent white noise amplification problem at low frequencies can be completely circumvented when the order of the desired beampattern is one.

On Phase Mode Selection in the Frequency-Invariant Beamformer for Near-Field mmWave Channel Characterization

On Phase Mode Selection in the Frequency-Invariant Beamformer for Near-Field mmWave Channel Characterization

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.

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.

Adaptive broadband frequency invariant beamforming using nulling-broadening and frequency constraints

The performance of conventional adaptive broadband beamforming algorithms is degraded because of rapidly moving interferences and look-direction errors. In this paper, a number of techniques are proposed to overcome this issue. By proposing an adaptive frequency invariant beamformer (FIB) that uses frequency constraints, we can obtain undistorted responses to broadband signal of interests (SOIs) and constant beamwidths over an entire frequency band without pre-steering delays. And the influence of look-direction errors can also be receded. Furthermore, a singular value decomposition of the frequency constraints is added to the proposed algorithm in view of releasing numerous degrees of freedom (DOFs), as this approach can simultaneously maintain the enhanced anti-interference performance and hold the undistorted response to SOIs. Finally, the nulling-broadening method involving the use of virtual interferences is adopted to construct a taper matrix and ensure robust performance versus rapidly moving interferences. This robust response to rapidly moving interferences and look-direction errors allows the proposed algorithm to achieve satisfactory broadband beamforming performance. Simulation results highlight the correctness and effectiveness of the proposed methods.

Insights Into the MMSE-Based Frequency-Invariant Beamformers for Uniform Circular Arrays

This letter provides new insights into the minimum mean square error (MMSE)-based frequency-invariant beamformer with the uniform circular array (FIB-UCA). By exploiting the symmetry of the UCA, we derive an explicit form of the white noise gain (WNG), the directivity factor (DF) and the MSE, and also derive the corresponding approximate expressions at low frequencies. We then investigate the impact of the look direction on the performance of the MMSE-based FIB-UCA. Interestingly, we further prove that for a UCA with the fixed aperture, the MMSE-based FIB with 2 <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> microphones achieves a similar WNG as that with 4 <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> microphones at low frequencies if the <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> th-order beampattern is steered to microphone angles. We also derive a unified explicit form of weighting vector and reveal that the MMSE-based FIB-UCA reduces to Jacobi-Anger expansion-based differential beamformer with the UCA at low frequencies.

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.

Frequency-Invariant Beam Steering Based on FIR-Filters

Conventional narrowband beamforming methods are not effective for wideband antenna arrays. Radiation pattern of such array has various beam width over the frequency. Existing frequency-invariant beamforming methods using finite impulse response filters (FIR-filters) can solve this problem. FIR-filters provide frequency response required for frequency-independent properties of the array radiation pattern but their common feature is a high computational complexity. In this paper, we present the fixed frequency-invariant beamforming method that does not require minimizing functions or the Fourier transform. The beam steering is achieved by using FIR-filters with different slope of the frequency response. The proposed approach is based on the FIR-filters synthesis with special frequency characteristics. The numerical results compared with the existing frequency-invariant beamforming method are shown.

Frequency Invariant Beamforming for a Small-Sized Bi-Cone Acoustic Vector-Sensor Array.

In this work, we design a small-sized bi-cone acoustic vector-sensor array (BCAVSA) and propose a frequency invariant beamforming method for the BCAVSA, inspired by the Ormia ochracea’s coupling ears and harmonic nesting. First, we design a BCAVSA using several sets of cylindrical acoustic vector-sensor arrays (AVSAs), which are used as a guide to construct the constant beamwidth beamformer. Due to the mechanical coupling system of the Ormia ochracea’s two ears, the phase and amplitude differences of acoustic signals at the bilateral tympanal membranes are magnified. To obtain a virtual BCAVSA with larger interelement distances, we then extend the coupling magnified system into the BCAVSA by deriving the expression of the coupling magnified matrix for the BCAVSA and providing the selecting method of coupled parameters for fitting the underwater signal frequency. Finally, the frequency invariant beamforming method is developed to acquire the constant beamwidth pattern in the three-dimensional plane by deriving several sets of the frequency weighted coefficients for the different cylindrical AVSAs. Simulation results show that this method achieves a narrower mainlobe width compared to the original BCAVSA. This method has lower sidelobes and a narrower mainlobe width compared to the coupling magnified bi-cone pressure sensor array.

Incoherent Synthesis of Sparse Arrays for Frequency-Invariant Beamforming

Frequency-invariant beamformers are used to prevent signal waveform distortions in real world applications like audio, underwater acoustics, and radar. Most of existing methods assume uniform arrays, and only few consider sparse designs, which may lead to higher performance in terms of robustness and directivity factor. We propose an incoherent approach that first determines for each frequency bin a sparse set of sensors positions. Subsequently, by using tools of dimensionality reduction and clustering, these selections are merged together yielding the optimal sensors on a sparse array layout. We present design examples of sparse linear and planar superdirective array designs. We show that the proposed incoherent sparse design obtains superior performance in terms of white noise gain, directivity factor, and computational load compared to a uniform array design and compared to a coherent sparse approach, where the sensors’ locations and the beamformer coefficients are optimized simultaneously for all frequencies.

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.

Empirical Study of Angular–Temporal Spectra in a Reverberation Chamber

The previous experimental studies on angular spectra of reverberation chambers (RCs) were conducted by either rotating a directive horn antenna (which suffers from low resolution) or using conventional beamforming (whose beam varies with frequency), resulting in inaccurate characterizations of angular statistics of the broadband channel in the RC. To avoid this problem, we resort to a frequency-invariant beamformer for the evaluation of the angular–temporal spectra of the RC. Contrary to the previous studies, it is found that modest loading does not increase the statistical anisotropy in the delay domain (after the ramp-up stage of the power delay profile) as compared with the unloaded case, implying that modest loading does not alter RC’s angular statistics for over-the-air testing.

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.

A Flexible Method for Designing Frequency-Invariant Beamformers With Circular Sensor Arrays

This paper proposes a flexible method for designing frequency-invariant (FI) beamformers, which is suitable for circular arrays mounted on acoustically transparent and infinite rigid cylinders. The closed-form weighting vectors at each frequency are accurately derived by approximating to a general complex-weighted desired beampattern from the minimum mean square error perspective. It is found that these obtained weighting vectors are all closed-form functions of the desired complex weighting vectors. A multiple-constraint optimization problem is formulated, and its solution gives the optimal desired weighting vector subject to different constraints. Then, the weighting vectors at other frequencies can be straightforwardly computed based on the previously derived analytical functions. The proposed method provides flexible schemes to design different FI beamformers for circular sensor arrays. The good performance of the proposed method is demonstrated using simulations and experimental results.

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.

Reference Beam Pattern Design for Frequency Invariant Beamforming Based on Fast Fourier Transform.

In the field of fast Fourier transform (FFT)-based frequency invariant beamforming (FIB), there is still an unsolved problem. That is the selection of the reference beam to make the designed wideband pattern frequency invariant (FI) over a given frequency range. This problem is studied in this paper. The research shows that for a given array, the selection of the reference beam pattern is determined by the number of sensors and the ratio of the highest frequency to the lowest frequency of the signal (RHL). The length of the weight vector corresponding to a given reference beam pattern depends on the reference frequency. In addition, the upper bound of the weight length to ensure the FI property over the whole frequency band of interest is also given. When the constraints are added to the reference beam, it does not affect the FI property of the designed wideband beam as long as the symmetry of the reference beam is ensured. Based on this conclusion, a scheme for reference beam design is proposed.