Broadband Beamformer Design Research Articles

An active set approximation method for the design of broadband beam former

The microphone array beam former design problem in far-field environment is to design the coefficients of FIR filters such that the maximum error between the actual response and the desired response in a specified domain is minimized. The specified domain is always very large and this problem is a large scale optimization problem. In general the complexity of the optimal method for solving this problem is very expensive. In this paper, we apply an active set approximation method to solve this problem. First, we transform this problem into a semi-infinite programming problem. Next, we find an approximate active set such that the scale of the problem can be mostly reduced. Then, this problem can be solved efficiently and the approximate solution is very close to the optimal solution. Numerical experiments are illustrated to show the efficiency of the proposed method.

Robust broadband beamformer design for noise reduction and dereverberation

In this paper, we investigate robust design methods for broadband beamformers in reverberant environments. In the design formulation room reverberation as well as robustness to amplitude and phase mismatches in the microphones have been included. Particularly, the direct path and the reflections are separated in the design such that there is a penalty on the reflective part. This approach is different from the commonly studied problem of dereverberation of a single point source as the investigated design is made over a region in space. A single point derverberation is not a very practical approach due to a high sensitivity to position changes. Thus in order to obtain more practical microphone array designs, we study methods that optimize performance over areas in space. The design problem has been formulated in four different ways; (i) using direct path only representing a traditional beamformer design method, (ii) using a robust design method which considers robustness against the microphone characteristics (gain and phase) by optimizing the mean performance, (iii) by including room impulse response in the design and finally, (iv) using both robustness and room impulse response in the design. Simulation results show that robust direct path based beamformer can achieve approximately the same performance as including room response in the design in many reverberation environments. The proposed method provides robustness over larger variations in the reverberation environment. This means that the robust direct path based method which is based on mean variations in gain and phase can be used in low- to medium ($$T_{60}$$ = 100–300 ms) reverberant environments with good result.

A Chance-Constrained Programming Approach to the Design of Robust Broadband Beamformers With Microphone Mismatches

This paper addresses the design of robust broadband beamformers by taking into account the prior knowledge of uncertain means and variances of microphone mismatches. A chance-constrained programming CCP based design approach is proposed, which is a generic extension of the existing worst case mean performance optimization WMPO based counterpart by using a chance-constrained stopband level constraint instead of a hard one. The original CCP-based design problem, however, is NP-hard to solve. To deal with the problem, a relaxed convex optimization formulation is developed by exploiting the distribution property of array response distortion due to microphone mismatches. The relation between the proposed CCP- and WMPO-based design approaches is analyzed theoretically. It shows that there always exists a chance constraint probability under which the CCP-based design is comparable to the WMPO-based one, which provides an insight into the overconservatism problem in the WMPO-based design from the perspective of the CCP-based framework. Moreover, a criterion on the setting of the chance constraint probability is also derived for the proposed design approach in order to remedy the overconservatism problem. Simulation and real experimental results are presented to show the superior performance of the proposed design approach compared with its state-of-the-art counterparts.

Constrained minimum power combination for broadband beamformer design in the STFT domain

Beamforming using sensor array is widely used in spatial signal processing since it offers better spatial focusing capability than single sensor. However, in practical applications for broadband signal, there always exists a trade-off issue between the directivity capability of an array and its robustness on system errors. In this paper, in order to combine merits of different beamformers instead of trade-off their performances, we propose a constrained minimum-power combination method. We firstly analyze two optimal beamformers that maximize Directivity Factor (DF) and White Noise Gain (WNG) respectively. Then we propose a non-linear combination method, which automatically selects the best beamformer that has the minimum output power, so as to control the unwanted white noise amplification and keep the maximum DF if possible. Two solutions to the proposed combination strategy are given. They do not need to determine the correct trade-off factor used in linear combination method, and avoid challenge estimations on noise and target statistics required in adaptive beamforming. The performance of the proposed beamformer is evaluated in ideal noise fields and complicated noise fields respectively. It is shown that the proposed beamformer integrates merits of different beamformers. It always achieves the best speech quality and biggest noise reduction compared to other popular beamformers.

Design of Robust Broadband Beamformers Using Worst-Case Performance Optimization: A Semidefinite Programming Approach

Broadband beamformers are known sensitive to microphone mismatches, especially for small-sized arrays. To address the problem, the worst-case performance optimization WCPO criterion has been used to design robust broadband beamformers. Recently, a circular-model-based design using the WCPO criterion was proposed, which has been shown superior to the existing second-order cone programming SOCP based design also using the WCPO criterion. In this paper, however, we find that the circular-model-based design is sensitive to stopband level constraint. If the constraint is chosen tightly, it may become less robust and degrades even worse than the SOCP-based design. To achieve a better tradeoff between robustness and beamforming performance, we propose a semidefinite programming based approach for robust broadband beamformer design using the WCPO criterion. The relations of the proposed design to its existing counterparts are also theoretically analyzed. In particular, we show that the proposed design is less conservative than the SOCP-based design, and is also less sensitive to the setting of stopband level constraint and more robust when compared to the circular model-based design. Another contribution of this paper is that the relation between the SOCP- and the circular-model-based designs is theoretically revealed. Simulation results and real experimental results are presented to show the superior of the proposed design over its existing counterparts.

Design of Minimax Broadband Beamformers that are Robust to Microphone Gain, Phase, and Position Errors

Broadband beamformers with small-size microphone arrays are known to be highly sensitive to microphone imperfections. A new method for the design of minimax broadband beamformers that are robust to microphone gain, phase, and position errors is proposed. In the method, the maximum variations in the microphone errors are used in formulating a convex optimization problem where the worst-case passband error is minimized under the constraint that the worst-case stopband error is below a prescribed level. To include the microphone imperfections in the optimization problem, we developed a suitable model that incorporates the variations due to the microphone errors and at the same time is efficient to compute. An important advantage of the proposed method is the availability of corresponding worst-case passband - and stopband-error bounds for the beamformer that has been designed; a second advantage is that it does not require the probability distributions of microphone errors. We then describe a two-phase method where the proposed method is used in the first phase to derive the passband and stopband error constraints for solving an optimization problem in the second phase where the white noise gain (WNG) of the beamformer is maximized. In our experiments, we compare beamformers designed using the proposed method, the two-phase method and a modified version of a competing method. Experimental results show that beamformers designed using the proposed method have much better performance than those of the modified competing method and comparable performance with those of the two-phase method; however, unlike the two-phase method, the proposed method provides the additional guarantee that the errors will always lie within the worst-case error bounds.

Design of minimax robust broadband beamformers with optimized microphone positions

A new method for the design of robust minimax far-field broadband beamformers with optimized microphone positions is proposed. The method is formulated as an iterative optimization problem where the maximum passband1 magnitude response error is minimized and the microphone positions are optimized while ensuring that the minimum stopband attenuation is above a prescribed level. To maintain robustness, we constrain a sensitivity parameter, namely, the white noise gain, to be above prescribed levels across the frequency band. An additional feature of the method, which is quite useful in certain applications, is that it provides the capability of constraining the gain in the transition band to always lie below the maximum gain in the passband. Performance comparisons with existing methods show that the optimization of the microphone positions results in beamformers with superior performance.

The design of multi-dimensional acoustic beamformers via window functions

The design of broadband beamformers can be formulated as a semi-infinite programming optimization problem, where the coefficients of the filters are determined such that the actual response of the microphone array is near a given desired response. This problem can be solved by existing optimization solvers after the discretization of the infinite constraints. However, this problem will become large-scale and it's expensive to find the optimal solution, as the discretization points grow and the filter length increases. In this paper, we propose a fast method based on the window functions. First, we formulate a simplified optimization problem to find the limit of the cost function values as the filter length is sufficiently long. Hence, the optimal frequency response vector is obtained and the corresponding filter coefficients can be calculated. Second, we apply the window method to truncate the limiting filter to obtain a finite FIR filters. The performance as well as the computational complexity are investigated to demonstrate the effectiveness and efficiency of the proposed method.

L -infinity norm design of linear-phase robust broadband beamformers using constrained optimization

A new method for the design of linear-phase robust far-field broadband beamformers using constrained optimization is proposed. In the method, the maximum passband ripple and minimum stopband attenuation are ensured to be within prescribed levels, while at the same time maintaining a good linear-phase characteristic at a prescribed group delay in the passband. Since the beamformer is intended primarily for small-sized microphone arrays where the microphone spacing is small relative to the wavelength at low frequencies, the beamformer can become highly sensitive to spatial white noise and array imperfections if a direct minimization of the error is performed. Therefore, to limit the sensitivity of the beamformer the optimization is carried out by constraining a sensitivity parameter, namely, the white noise gain (WNG) to be above prescribed levels across the frequency band. Two novel design variants have been developed. The first variant is formulated as a convex optimization problem where the maximum error in the passband is minimized, while the second variant is formulated as an iterative optimization problem and has the advantage of significantly improving the linear-phase characteristics of the beamformer under any prescribed group delay or linear-array configuration. In the second variant, the passband group-delay deviation is minimized while ensuring that the maximum passband ripple and stopband attenuation are within prescribed levels. To reduce the computational effort in carrying out the optimization, a nonuniform variable sampling approach over the frequency and angular dimensions is used to compute the required parameters. Experiment results show that beamformers designed using the proposed methods have much smaller passband group-delay deviation for similar passband ripple and stopband attenuation than a modified version of an existing method.

Design of Robust Broadband Beamformers With Discrete Coefficients and Least Squared Criterion

This brief investigates the performance of robust and nonrobust broadband beamformers with least squares and discrete coefficients to achieve low complexity and efficient hardware implementation. The broadband beamformer coefficients are expressed as the sum of power-of-two terms with a restriction on the total number of power-of-two terms for the beamformer coefficients. An iterative algorithm is employed to reduce the number of nonzero coefficients and, thereby, multipliers in both the robust and nonrobust beamformers. A quantization scheme in combination with a random search is then applied to efficiently distribute the power-of-two terms for the beamformer coefficients. Design examples show that the number of nonzero coefficients for the beamformers can be significantly reduced without a significant degradation in the integral squared error. In addition, robust beamformers are shown to be less sensitive to nonzero coefficient reduction and quantization than nonrobust beamformers.

Design of Steerable Spherical Broadband Beamformers With Flexible Sensor Configurations

In broadband beamformer applications with dynamically moving sources, it can be important to have a simple mechanism to steer the main-beam. It can also be desirable if the beampattern of the beamformer is invariant to the look direction. A number of design methods for such beamformers, based on the spherical harmonic transform, have been reported in the literature. However, these methods require the sensor positions to satisfy a certain condition which may conflict with practical considerations. This paper proposes a design method which obviates this restriction thus allowing for spherical arrays with arbitrary sensor configurations. Moreover, for a comparable level of performance and computational complexity to the existing spherical harmonic beamformers, the proposed beamformer requires fewer sensors. The trade-off is that the design of the beamformer now depends on the sensor positions. Other considerations, such as the effects of array mis-orientation and robustness, are also discussed in this paper and illustrated by design examples.

Indoor beamformer design using room simulators

Broadband microphone array provides an important means of hands-free speech acquisition via spatial beamforming techniques. There are many different approaches for the design of near-field broadband beamformers. One method is based on the wave propagation model using a direct path transfer function. However, as signals are corrupted by different interfering noise, room reverberation plays a particular important role for indoor applications even if there is no another speaker around. Image-source method is one simple but effective approach for room acoustic simulation. It is also possible to employ a wave-based model for the simulation. In this paper, we will study different room simulators and employ them for the design of indoor beamformers.

Design of broadband beamformers with low complexity

Abstract In this article, we consider the design of broadband beamformers with low complexity. In fact, the design problem is multi-objective in nature, trading off between speech distortion and noise suppression. Finding a balance between these two objectives is important in order to achieve a desired sound quality. These measures are introduced as the objectives here. The design can then be obtained via a bi-objective integer programming problem, where the coefficients of the filters are expressed as sums of signed powers-of-two terms. We study two different integer spaces and penalty functions for solving the problem. Then, an algorithm based on a discrete filled function is developed for finding the optimal design. In order to illustrate the effectiveness of the algorithm, real data is used and two broadband beamformers are demonstrated.

A Two-Stage Method for the Design of Near-Field Broadband Beamformer

In this paper, the design of near-field broadband beamformer is considered. The idea is to design the beamformer filter coefficients such that the error between the actual response and the desired response is minimized. This problem can be formulated as a minimax optimization problem and can be transformed into a linear semi-definite programming problem. Interior point algorithm can then be applied after discretization. However, if the dimension of the problem is high, the number of discrete points of the specified region and consequently the dimension of the resultant constraint matrix is high. This high dimension provides a difficulty when directly employing interior point algorithm. Since it results in a large size problem, this problem is considered in this paper. To reduce the computational complexity and memory usage, a two-stage method has been proposed. In the first stage, by using an infinite length filter formulation, an infinite length limit for the original problem is obtained. Then, based on that limit, a reduced problem with a finite filter length is found. The computational complexity of this method is derived. It provides a significant reduction compared to a direct solution of the original problem. It is shown that as the filter length increases, the approximative problem comes close to the optimal infinite solution. Furthermore, we demonstrate with examples the close correspondence between proposed design and optimal design. Also the superiority of the suggested method is illustrated in a three-dimensional beamformer design, where the direct method can only design broadband beamformer of very short filter lengths.

A Computationally Efficient Procedure for the Design of Robust Broadband Beamformers

Broadband beamformers using superdirective arrays are sensitive to errors in the sensor characteristics. Recently, a method that considers the probability distributions of errors to synthesize a robust optimal beamformer has been proposed. Unfortunately, the entire synthesis process may become unacceptably long due to the computational burden of the cost function. Here, we present an alternative procedure for computing the same cost function, drastically reducing such a problem.

Design of Robust Broadband Beamformers With Passband Shaping Characteristics Using Tikhonov Regularization

Broadband beamformers are known to be highly sensitive to the errors in microphone array characteristics, especially for small-sized microphone arrays. This paper proposes an algorithm for the design of robust broadband beamformers with passband shaping characteristics using Tikhonov regularization method by taking into account the statistics of the microphone characteristics. To facilitate the derivation, a weighted least squares based broadband beamforming algorithm with passband shaping characteristics is also proposed, while keeping a minimal stopband level. Unlike the existing criteria for broadband beamforming for microphone arrays, the proposed approaches fully exploit the degrees of freedom in weighting functions of the criteria used to design broadband beamformers. By doing so, efficient and flexible broadband beamforming can be achieved with the desired passband shaping characteristics. In addition, this paper also shows that the robust broadband beamforming using the statistics of the microphone characteristics recently proposed (Doclo and Moonen, 2003 and 2007) belongs to the class of white noise gain constraint-based techniques, which gives an insight into the nature of the state-of-the-art approach. The performance of the proposed beamforming algorithms is illustrated by the design examples and is compared with the existing approaches for microphone arrays.

Generalized Broadband Beamforming Using a Modal Subspace Decomposition

We propose a new broadband beamformer design technique which produces an optimal receiver beam pattern for any set of field measurements in space and time. The modal subspace decomposition (MSD) technique is based on projecting a desired pattern into the subspace of patterns achievable by a particular set of space-time sampling positions. This projection is the optimal achievable pattern in the sense that it minimizes the mean-squared error (MSE) between the desired and actual patterns. The main advantage of the technique is versatility as it can be applied to both sparse and dense arrays, nonuniform and asynchronous time sampling, and dynamic arrays where sensors can move throughout space. It can also be applied to any beam pattern type, including frequency-invariant and spot pattern designs. A simple extension to the technique is presented for oversampled arrays, which allows high-resolution beamforming whilst carefully controlling input energy and error sensitivity.

Near-field broadband beamformer design via multidimensional semi-infinite linear programming techniques

Broadband microphone arrays has important applications such as hands-free mobile telephony, voice interface to personal computers and video conference equipment. This problem can be tackled in different ways. In this paper, a general broadband beamformer design problem is considered. The problem is posed as a Chebyshev minimax problem. Using the l/sub 1/-norm measure or the real rotation theorem, we show that it can be converted into a semi-infinite linear programming problem. A numerical scheme using a set of adaptive grids is applied. The scheme is proven to be convergent when a certain grid refinement is used. The method can be applied to the design of multidimensional digital finite-impulse response (FIR) filters with arbitrarily specified amplitude and phase.

Design of broadband beamformers robust against gain and phase errors in the microphone array characteristics

Fixed broadband beamformers using small-size microphone arrays are known to be highly sensitive to errors in the microphone array characteristics. The paper describes two design procedures for designing broadband beamformers with an arbitrary spatial directivity pattern, which are robust against gain and phase errors in the microphone array characteristics. The first design procedure optimizes the mean performance of the broadband beamformer and requires knowledge of the gain and the phase probability density functions, whereas the second design procedure optimizes the worst-case performance by using a minimax criterion. Simulations with a small-size microphone array show the performance improvement that can be obtained by using a robust broadband beamformer design procedure.

Design of far-field and near-field broadband beamformers using eigenfilters

This paper discusses two novel non-iterative design procedures based on eigenfilters for designing broadband beamformers with an arbitrary spatial directivity pattern for an arbitrary microphone configuration. In the conventional eigenfilter technique a reference frequency-angle point is required, whereas in the eigenfilter technique based on a TLS (total least squares) error criterion, no reference point is required. It is shown how to design broadband beamformers in the far-field, near-field and mixed near-field far-field of the microphone array. Both eigenfilter techniques are compared with other broadband beamformer design procedures (least-squares, maximum energy array, non-linear criterion). It will be shown by simulations that among the considered non-iterative design procedures the TLS eigenfilter technique has the best performance, i.e. best resembling the performance of the non-linear design procedure but having a significantly lower computational complexity.