White Noise Gain Research Articles

On direct optimization in mode space for robust supergain beamforming of circular array mounted on a finite cylinder

A direct optimization method in mode space for robust supergain beamforming of circular array mounted on a finite rigid cylinder is presented. According to the concept of eigen-decomposition, the beam pattern is decomposed into a series of eigen-beams weighted by modal coefficients. The modal cross spectral matrix in isotropic noise field is calculated from sound scattering theory based on boundary element method(BEM). This beamforming method gives the most suitable modal coefficient vector directly under the related constraint conditions via second-order cone programming, so that there is no need to transform indirectly from the weighting vector in sensor space which is essential in the modal robust supergain beamforming method proposed before. The results of simulation show that the direct modal beamforming method in this paper can not only improve robustness using the white noise gain constraint, but also change the mode orders to provide trade-off between array gain and robustness in low frequencies. Beam performance measures such as sidelobe level can be optimized in addition to array gain, and in this way more effective schemes for designing practical robust supergain beamformers can be developed.

Robust Minimum Sidelobe Beamforming for Spherical Microphone Arrays

A robust minimum sidelobe beamforming approach based on the spherical harmonics framework for spherical microphone arrays is proposed. It minimizes the peaks of sidelobes while keeping the distortionless response in the look direction and maintaining the mainlobe width. A white noise gain constraint is also derived and employed to improve the robustness against array errors. The resulting beamformer can provide optimal tradeoff between the sidelobe level, the beamwidth and robustness, so it could be more practical than the existing spherical array Dolph-Chebyshev modal beamformer in the presence of array errors. The optimal modal beamforming problem is formulated as a tractable convex second-order cone programming program, which is more efficient than conventional element-space based approaches, since the dimension of array weight vectors can be significantly decreased by using the properties of spherical harmonics and Legendre polynomials. For the purpose of performance comparison, we also formulate current robust modal beamformers as equivalent optimization problems based on the proposed array model. Numerical results show the high flexibility and efficiency of the proposed beamforming approach.

Beamforming for directional sources: Additional estimator and evaluation of performance under different acoustic scenarios

Beamforming is done with an array of sensors to achieve a directional or spatially-specific response by using a model of the arriving wavefront. Conventionally, a plane wave or point source model is used and this can cause decreased array gain or even total breakdown of beamforming when the source is directional. To avoid this, the authors proposed in recent work an alternative beamforming method which defines a set of "sub-beamformers," each designed to respond to a different spatial mode of the source. The outputs of the individual sub-beamformers are combined in a weighted sum to give an overall output of better quality than that of a monopole beamformer. This paper extends the previous work by introducing an additional estimator for the weighted sum and by presenting simulation results to demonstrate the relative performance of the proposed method and the different estimators for a directional source in the presence of diffuse noise, reverberation, and an interfering source. Gain optimization subject to a constraint on the white-noise gain with the proposed beamforming method is also introduced. Generally, when beamforming on directional sources, the proposed method outperforms beamforming with a point source model when the input signal-to-noise ratio (SNR) is 0 dB or higher.

Optimal spherical array modal beamformer in the presence of array errors.

Spherical array modal beamforming has become an attractive technique, since three-dimensional beampatten synthesis is more flexible than other array geometries, and the modal beamforming can be performed using the elegant spherical harmonics framework. However, the performance of a modal beamformer in practical situations is known to degrade in the presence of array errors caused by non-perfect spatial sampling, sensor sensitivity and phase variations, and sensor self-noise. Although the white noise gain constraint has been widely used to control the robustness of modal beamformers, it is not clear how to properly choose the constrained parameter set based on the known range of errors. In this paper, a worst-case performance optimization approach is formulated for the spherical array modal beamformer to improve its robustness against the above mentioned errors occurring in practice; thus the optimum performance can be obtained based on the known maximum level of errors. The robust optimal modal beamforming problem is reformulated as a convex optimization within the spherical harmonics framework, which can be efficiently solved by using second order cone programming.

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.

Large Code Set for Double User Capacity and Low PAPR Level in Multicarrier Systems

In this paper, a new large spreading code set with a uniform low cross-correlation is proposed. The proposed code set is capable of (1) increasing the number of assigned user (capacity) in a multicarrier code division multiple access (MC-CDMA) system and (2) reducing the peak-to-average power ratio (PAPR) of an orthogonal frequency division multiplexing (OFDM) system. In this paper, we derive a new code set and present an example to demonstrate performance improvements of OFDM and MC-CDMA systems. Our proposed code set with code length of N has K = 2N + 1 number of codes for supporting up to (2N + 1) users and exhibits lower cross correlation properties compared to the existing spreading code sets. Our results with subcarrier N = 16 confirm that the proposed code set outperforms the current pseudo-orthogonal carrier interferometry (POCI) code set with gain of 5 dB at bit-error-rate (BER) level of 10-4 in the additive white Gaussian noise (AWGN) channel and gain of more than 3.6 dB in a multipath fading channel.

The Spherical-Shell Microphone Array

Spherical microphone arrays have been recently studied for a wide range of applications. In particular, microphones arranged around an open or virtual sphere are useful in scanning microphone arrays for sound field analysis. However, open-sphere spherical arrays have been shown to have poor robustness at frequencies related to the zeros of the spherical Bessel functions. This paper presents a framework for the analysis of array robustness using the condition number of a given matrix, and then proposes several robust array configurations. In particular, a dual-sphere configuration previously presented which uses twice as many microphones compared to a single-sphere configuration is analyzed. This paper then shows that high robustness can be achieved without increasing the number of microphones by arranging the microphones in the volume of a spherical shell. Another simpler configuration employs a single sphere and an additional microphone at the sphere center, showing improved robustness at the low-frequency range. Finally, the white-noise gain of the arrays is investigated verifying that improved white-noise gain is associated with lower matrix condition number.

Open-Loop Adaptive Filter for Power Electronics Applications

An open-loop adaptive filter, which can be used in various power electronics control systems, is presented. The purpose of the filter is to give as fast a step response as possible while retaining good noise canceling properties. The proposed filter consists of a first-order infinite-impulse-response filter, the coefficients of which are adapted according to the changes in the input signal. In particular, good performance is obtained if the filter must suppress a low-frequency narrow-band noise. The performance of the filter is verified through step response simulations and with a laboratory prototype of a welding machine. The control system of the prototype is implemented on a field-programmable gate array. The proposed filter is compared to two optimum linear filters: (1) the moving average filter and (2) a filter whose step response and white noise gain are both optimized. The step response simulations show that, with the proposed filter, the step response time can be considerably decreased compared to these filters. The prototype also shows that, when using the proposed filter, the step response of the welding current can be made better than that of the original analog control.

Decoupled Beamforming and Noise Cancellation

The enhancement of noise-corrupted speech acquired by microphones is indispensable to the functioning of a wide variety of digital signal processing algorithms. Many existing products are equipped with steerable, stand-alone fixed beamformers which provide moderate levels of directivity. Moreover, many applications have long employed the classical adaptive noise canceller configuration with a reference sensor near the noise source to cancel unwanted noise. In this paper, the cascading of stand-alone beamformers with back-end adaptive noise cancellers is studied. A decoupled model for signal enhancement using front-end beamformers and cascaded noise cancellers is presented. The inter-operation of the beamforming and noise canceling units is studied by defining the signal-to-interference ratio (SIR) gain, directivity index, and white noise gain offered by the beamforming and noise cancelling components. The performance of decoupled beamformer-noise canceller structures is evaluated using experimental measurements. An experimental procedure for evaluating output SIR is presented. Results reveal SIR improvements of up to 27 dB, and are compared to those stemming from conventional adaptive beamformers

Superdirective Beamforming Robust Against Microphone Mismatch

Fixed superdirective beamformers using small-sized microphone arrays are known to be highly sensitive to errors in the assumed microphone array characteristics (gain, phase, position). This paper discusses the design of robust superdirective beamformers by taking into account the statistics of the microphone characteristics. Different design procedures are considered: applying a white noise gain constraint, trading off the mean noise and distortion energy, minimizing the mean deviation from the desired superdirective directivity pattern, and maximizing the mean or the worst case directivity factor. When computational complexity is not an issue, maximizing the mean or the worst case directivity factor is the preferred design procedure. In addition, it is shown how to determine a suitable parameter range for the other design procedures such that both a high directivity and a high level of robustness are obtained

Robust supergain beamforming for circular array via second-order cone programming

The phenomenon of supergain for a circular array and its robust beamforming are presented. The coplanar superdirective array gain of the circular array, although it is not so extreme as an endfire line array, outperforms a lot over that of a conventional delay-and-sum beamformer in isotropic noise fields when the inter-element spacings are much smaller than one-half wavelength. However, optimum beamforming algorithms can be extremely sensitive to slight errors in array characteristics. The performance are known to degrade significantly if some of underlying assumptions on the sensor array is violated. Therefore, white noise gain constraint is used to improve the robustness of the supergain beamformer against random errors. We show that the design of the weight vector of robust supergain beamformer can be reformulated as a form of second-order cone programming and resolved efficiently via the well-established interior point method. Results of computer simulation for a 24-element circular array confirm satisfactory performance of the approach proposed in this paper.

Approaches to robustness

The term robustness in signal processing applications usually refers to approaches that are not degraded significantly when the assumptions that were invoked in defining the processing algorithm are no longer valid. Highly tuned algorithms that fall apart in real-world conditions are useless. The classic example is super-directive arrays of closely spaced elements. The very narrow beams and high directivity could be predicted under ideal conditions, could not be achieved under realistic conditions of amplitude, phase and position errors. The robust design tries to take into account the real environment as part of the optimization problem. This problem led to the introduction of the white noise gain constraint and diagonal loading in adaptive beam forming. Multiple linear constraints have been introduced in pursuit of robustness. Sonar systems such as towed arrays operate in less than ideal conditions, making robustness a concern. A special problem in sonar systems is failed array elements. This leads to severe degradation in beam patterns and bearing response patterns. Another robustness issue arises in matched field processing that uses an acoustic propagation model in the beamforming. Knowledge of the environmental parameters is usually limited. This paper reviews the various approaches to achieving robustness in sonar systems.

Robust adaptive beamforming for hydrodynamic self-noise rejection

White noise gain (WNG) is a metric widely used in the radar community for limiting the degree of adaptivity of an adaptive beamformer (ABF). Constraining adaptation of an ABF algorithm has two key effects: (1) protection against self-nulling associated with steering vector mismatch, and (2) limitation of white noise gain associated with squinting a beampattern to place a null on a mainlobe interferer. Hydrodynamic self-noise, or cable strum, commonly manifests itself as a source of mainlobe interference for passive acoustic towed horizontal line arrays. Cable strum is the result of vibrations excited in the array body by vortex shedding in the presence of flow. Strum is particularly detrimental on beams near endfire and typically exhibits very high dynamic range. As such, it requires an aggressive adaptation strategy for its effective removal. In this work, a white noise gain constraint for the rejection of cable strum is derived based on the scaled orthogonal projection approach of Cox et al. [IEEE Trans. ASSP 35 (1987)]. Tradeoffs involved in balancing strum rejection performance against mismatch-induced signal self-nulling are examined. [Sponsored by the Dept. of the Navy, under Air Force Contract No. F19628-00-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and not necessarily endorsed by the U.S. Air Force.]

Beamforming for a circular microphone array mounted on spherically shaped objects

This paper will discuss the concept of phase modes to generate a desired beam pattern for a circular microphone array mounted around a rigid sphere. The method will be described for arrays consisting of omnidirectional and dipole sensors. The sound diffraction caused by the sphere is taken into account. It will be seen that the method allows, with some restrictions, the design of a wide variety of broadband beam patterns for a given elevation which usually will be the plane of the array. The directivity index is used to characterize the three-dimensional behavior of the array. Simulations show the realization of different beam patterns, based on a 16-element circular array located at the equator of a sphere with radius 0.085 m. The frequency range of this array is from 300 Hz to 5 kHz. Especially at low frequencies, a very good combination of the directivity index and the white noise gain is achieved which cannot be realized with “conventional” beamforming for an array of similar dimensions. The simulations are verified by means of a measurement.

Initial evaluation of the dominant mode rejection beamformer

Long integration times are needed to give accurate cross-spectral matrix (CSM) estimates; however, integration times must be short enough so that the dynamic behavior of the noise described by the CSM is captured. The dominant mode rejection (DMR) beamformer, described by Owsley and Abraham, calculates adaptive weights based on a reduced rank (CSM) estimate, where the CSM estimate is formed with a subset of the eigenvalues containing the largest eigenvalues and their eigenvectors. Thus the integration time required to obtain this CSM estimate is reduced, as the largest eigenvalue/eigenvector pairs are estimated much more rapidly than the smaller ones. The purpose of this study was to determine whether performance comparable to a fully adaptive beamformer could be achieved with the DMR beamformer, without the potential penalties associated with requiring the increased integration time for the CSM. Data acquired with bottom-mounted horizontal line arrays, in both deep- and shallow-water environments, were used to test the dominant mode rejection beamformer performance. Appropriate DMR parameters were found using the deep-water data so that the DMR performance emulated the performance of a fully adaptive beamformer with a white-noise gain constraint. Additional processing performed on the shallow-water data showed that the DMR beamformer had equivalent performance to a robust fully adaptive beamformer, but with much less output power bias at small integration times.

Optimization of quiescent response in partially adaptive beamformers

The adaptive weights utilize only a fraction of the available degrees of freedom in partially adaptive LCMV (linearly constrained minimum-invariance) beamformers. Two procedures are presented for incorporating unused degrees of freedom into the beamformer's quiescent response. In one case the quiescent response is chosen to approximate a desired quiescent response, and in the other case it is chosen to maximize the average interference cancellation over a set of likely interference scenarios. A sidelobe-canceling interpretation indicates that maximization of average interference cancellation results in quiescent weights that optimize the sum of the white noise gain and the error between the quiescent response and the class of responses which the adaptive branch can synthesize. Simulations illustrate the utility of the methods. >

Practical supergain

The problem considered is that of designing endfire line array shadings which provide a useful amount of supergain without extreme sensitivity to random errors. Optimum shading weights are obtained subject to a constraint on the gain against uncorrelated white noise. The results of optimum array gain versus white noise gain constraint are presented parametrically for arrays of different interelement spacings, and different noise fields. Results are presented for spherically and cylindrically isotropic noise, and other wavenumber limited noise fields, used in modeling ocean ambient noise. It is found that nearly optimum performance can be obtained in a simple delay and sum beamformer by shading to reduce sidelobes and modest oversteering to reduce mainlohe width without too large a reduction in mainlobe sensitivity.