White Noise Gain Research Articles

Overcoming snapshot-deficient measurements with knowledge-aided approaches.

The use of knowledge-aided covariance is considered for processing underwater acoustic array data in snapshot-deficient scenarios. The knowledge-aided formalism is a technique that combines array data with a known covariance to produce an invertible estimate. For underwater acoustics, simulations of ambient noise provide the a priori covariance allowing degraded signals to be processed adaptively in situations where the sample covariance matrix is rank-deficient. The method is demonstrated for matched field processing using the 21 element array event S5 from the SWellEx-96 experiment. With five snapshots, the knowledge-aided approach significantly reduces localization ambiguity compared to the adaptive white noise gain constraint processor.

Very low frequency three-dimensional beamforming for a miniaturized aperture acoustic vector sensor array.

In this Letter, a three-dimensional beamforming method is proposed for a miniaturized aperture acoustic vector sensor (AVS) array. This method extracts the multipole modes using the AVS array, and then synthesizes the desired beam pattern according to the relation between multipole modes and spherical harmonics. Compared to the spherical harmonics decomposition of the sound field method, the proposed method achieves comparable array gain with higher white noise gain when the distance of adjacent elements d and the wavelength λ satisfy 0.01≤d/λ≤0.2. Experimental results also demonstrate the effectiveness of the proposed three-dimensional beamforming method at very low frequency.

Fundamental Approaches to Robust Differential Beamforming With High Directivity Factors

Differential beamforming, which measures the spatial derivatives of the acoustic pressure field, can be used in a wide range of small devices that require high-fidelity sound and speech acquisition as it can achieve frequency-invariant spatial responses with high directivity factors (DFs). Since a differential process is inherently sensitive to sensors' self noise and other array imperfections, the most challenging problem in the design of any differential beamformer is how to achieve the maximum possible DF while maintaining a proper level of robustness for practical usage. While significant efforts have been made on this topic, the problem remains unsolved and further study is indispensable. This paper is devoted to dealing with this challenging problem. It presents a study on theory and methods to achieve the optimal and fundamental compromise between the white noise gain (WNG), which quantifies how robust is the beamformer, and the DF in differential beamforming. The major contributions of this work are as follows. 1) We show and prove that any null constrained fixed beamformer can be decomposed as the sum of two orthogonal filters, i.e., the maximum WNG (MWNG) beamformer and a reduced-rank one. Based on this decomposition, we develop three kinds of differential beamformers from the WNG perspective, which can achieve a flexible and optimal compromise between DF and WNG. 2) We show that a transformed null constrained beamformer can also be decomposed as the sum of two orthogonal filters, i.e., the transformed maximum DF (MDF) beamformer and another reduced-rank one. Based on this decomposition, we also develop three kinds of differential beamformers, which can obtain the desired level of DF while using the rest of the degrees of freedom to maximize the WNG. Simulations are performed to validate the theoretical analysis and developed differential beamformers.

Random Matrix Theory Predictions of Dominant Mode Rejection Beamformer Performance

Adaptive beamformers use a sensor covariance matrix estimated from data snapshots to mitigate directional interference and attenuate uncorrelated noise. Dominant mode rejection (DMR) is a variant of the classic minimum variance distortionless response (MVDR) algorithm that replaces the smallest eigenvalues in the covariance estimate by their average while leaving the largest ones unmodified. Since DMR does not invert the small eigenvalues of the sample covariance, it achieves a higher white noise gain than MVDR while still suppressing loud interferers, thereby yielding a higher signal-to-interference-plus-noise ratio (SINR). MVDR achieves an average output SINR within 3 dB of the SINR achievable with the true covariance with approximately twice as many snapshots as sensors. Prior empirical analyses showed that achieving the same SINR with DMR only requires approximately twice as many snapshots as interferers. This paper derives analytical models of white noise gain and interference leakage for DMR using random matrix theory (RMT), specifically the eigenvalue and eigenvector limiting results of the spiked covariance model. Applying the new analytical models confirms the empirical DMR snapshot requirement. To handle finite cases with a large number of interferers relative to the array size, this paper derives a modified spiked covariance model that improves the accuracy of RMT results, and hence DMR performance predictions.

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.

A Framework of Directional-Gain Beamforming and a White-Noise-Gain-Controlled Solution

It is well known that an adaptive beamformer can be decomposed as a fixed beamformer followed by a post filter. This decomposition gives a much flexible way to design robust adaptive beamformers with high array gain and consequently it has become a popular approach to speech enhancement. In such a framework, the most critical problem is to estimate the post filter, which is studied in this paper. We present a multistage approach to the design of the post filter, which consists of a primary beamformer, a secondary beamformer and several auxiliary beamformers. Since the designed post filter is a function of source incidence angle by nature, we call it a directional gain. To evaluate the directivity of the beamformers in computing the directional gain, we introduce a modified beampattern, which is a function of both the source incidence angle and the point-source-to-background-noise ratio (PBR). To validate the presented approach, we analyze the principles of the primary, secondary and auxiliary beamformers, and then present a way to design these beamformers under the constraint of minimum white-noise-gain (WNG). Finally, we evaluate the performance of the directional gain and compare it with the traditional beamformers. The results show that the proposed directional gain can help achieve higher directivity factors (DFs), and better point-source-noise attenuation.

Constant-Beamwidth Beamforming with Concentric Ring Arrays.

Designing beampatterns with constant beamwidth over a wide range of frequencies is useful in many applications in speech, radar, sonar and communication. In this paper, we design constant-beamwidth beamformers for concentric ring arrays. The proposed beamformers utilize the circular geometry to provide improved beamwidth consistency compared to beamformers which are designed for linear sensor arrays of the same order. In the proposed configuration, all sensors on each ring share the same weight value. This constraint significantly simplifies the beamformers and reduces the hardware and computational resources required in a physical setup. Furthermore, a theoretical justification of the beamforming method is provided. We demonstrate the advantages of the proposed beamformers compared to the one-dimensional configuration in terms of directivity index, white noise gain and sidelobe attenuation.

Designing a covariance matrix tapered beamformer that is universal over notch width

Universal adaptive beamformers (UABFs) combine classic adaptive beamformers with ensemble methods from machine learning to design new beamformers whose asymptotic performance rivals the performance of the best beamformer in a set. The UABF array weights blend a weighted average of the competing beamformer array weights based on past performance using the softmax function. This model averaging approach avoids a difficult model selection problem by hedging performance across a family of models rather than committing to a single model. This talk presents a UABF which blends covariance matrix tapers (CMT). The CMT technique creates broader notches in beampatterns for environments with moving interferers. Interferers moving with high bearing rates cross resolution cells faster and violate the stationarity solution of the minimum variance distortionless response (MVDR) beamformer. The performance of the CMT depends on the chosen notch width. The UABF adapts to find the blend of CMT notch widths that best balances the interferer output power with white noise gain. We present simulation results illustrating the benefits of the UABF approach for CMT in the presence of moving interferers. [Work supported by ONR 321US.]

Exploiting the directional coherence function for multichannel source extraction

The desired speech detector plays an important role for controlling the speech distortion in spatial filtering based speech enhancement algorithms. However, the conventional complex coherence(CC) based algorithms can only distinguish the coherent speech and diffuse noise. To improve the performance on the scenarios that both the coherent interference and diffuse noise are present, we propose a directional coherence function(DCF) based detector in this paper. Based on a pair of complementary filters which can suppress the diffuse noise and the coherent interference respectively, the DCF is computed as the normalized correlation between the filters’ outputs. Meanwhile, the filters are solved by convex programming method and satisfy the constraints on speech distortionless and white noise gain(WNG). Consequently, the value of DCF will be close to 1 only for the desired speech dominated time-frequency(T-F) bins and much smaller than 1 for the noise or interference dominated T-F bins. To extract the desired speech, the DCF based Desired Speech Presence Probability(DSPP) is used to control the adaptation in general sidelobe canceler(GSC), and subsequently used as the post-filtering weight. Systematical experiments on several scenarios show that the proposed algorithm achieves significantly and consistently better noise suppression performance than the narrowband direction-of-arrival(DOA) estimates based algorithms.

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.

Near-Field Superdirectivity: An Analytical Perspective

The gain achieved by a superdirective beamformer operating in a diffuse noise-field is significantly higher than the gain attainable with conventional delay-and-sum weights. A classical result states that for a compact linear array consisting of $N$ sensors which receives a plane-wave signal from the end-fire direction, the optimal superdirective gain approaches $N^2$ . It has been noted that in the near-field regime higher gains can be attained. The gain can increase, in theory, without bound for increasing wavelength or decreasing source-receiver distance. We aim to address the phenomenon of near-field superdirectivity in a comprehensive manner. We derive the optimal performance for the limiting case of an infinitesimal-aperture array receiving a spherical-wave signal. This is done with the aid of a sequence of linear transformations. The resulting gain expression is a polynomial, which depends on the number of sensors employed, the wavelength, and the source-receiver distance. The resulting gain curves are optimal and outperform weights corresponding to other superdirectivity methods. The practical case of a finite-aperture array is discussed. We present conditions for which the gain of such an array would approach that predicted by the theory of the infinitesimal case. The white noise gain (WNG) metric of robustness is shown to increase in the near-field regime.

On the Robustness of the Superdirective Beamformer

In microphone array beamforming, a high directional gain is always desired for acoustic noise and reverberation suppression; as a result, the superdirective beamformer has been of great interest in many applications. However, this beamformer is well known to be very sensitive to array imperfections. While much effort has been made to improve its robustness, it is still a major problem. This paper is essentially devoted to the study of the robustness of the superdirective beamformer and derivation of better ways to deal with this important issue. We first prove that any distortionless fixed beamformer can be written as the sum of two orthogonal beamformers, i.e., the sum of the classical delay-and-sum (DS) beamformer and a reduced-rank beamformer. Based on this property, different kinds of robust superdirective beamformers are then developed. We also show that the robust design problem can be transformed into a quadratic eigenvalue problem (QEP), which leads to a solution that achieves the maximum possible directivity factor (DF) while meets the white noise gain (WNG) constraint over a frequency band of interest.

On the Design of Differential Kronecker Product Beamformers

In this paper, we present a generalized approach for differential microphone array (DMA) beamforming in the short-time Fourier transform (STFT) domain. We propose a multistage beamforming approach, which considers a Kronecker product (KP) decomposition of the global beamformer into two independent sub-beamformers. We derive differential KP beamformers according to different criteria and analyze their performances, which are tuned by three design parameters. These parameters allow a high beamforming design flexibility; in particular, non-differential or non-KP beamformers may be obtained as special cases. Depending on the selection of parameters, we demonstrate a preferable performance with the new approach with respect to the white noise gain and directivity factor measures. In addition, we consider the task of speech enhancement. We show that differential KP beamformers perform better than non-differential and non-KP beamformers in terms of the quality and intelligibility of their respective time-domain enhanced signals, particularly in moderately reverberant environments.

Beamforming with Cube Microphone Arrays Via Kronecker Product Decompositions

Microphone arrays combined with beamforming have been widely used to solve many important acoustic problems in a wide range of applications. Much effort has been devoted in the literature to microphone array beamforming, among which the Kronecker product beamforming method developed recently has demonstrated some interesting properties. Generally, this method decomposes the global beamforming filter into a Kronecker product of a number of sub-beamforming filters, each of which corresponds to a virtual subarray and can be designed individually. This decomposition not only reduces significantly the number of beamforming coefficients, but also can be explored to improve the robustness and flexibility of beamforming. This paper extends Kronecker product beamforming from two-dimensional arrays into three-dimensional cube arrays. We consider two decompositions, i.e., fully and partially separable ones. The former decomposes the entire array into three linear subarrays while the latter decomposes the entire array into a linear subarray and a planar one. Then, for each case, we derive the Kronecker product maximum white noise gain beamformer, the Kronecker product approximate maximum directivity factor (DF) beamformer, the Kronecker product null-steering beamformer, and the Kronecker product iterative maximum DF beamformer. Simulation results demonstrate the properties and advantages of the proposed beamformers.

Evaluation of head-tracked binaural auralizations of speech signals generated with a virtual artificial head in anechoic and classroom environments

In order to realize binaural auralizations with head tracking, BRIRs of individual listeners are needed for different head orientations. In this contribution, a filter-and-sum beamformer, referred to as virtual artificial head (VAH), was used to synthesize the BRIRs. To this end, room impulse responses were first measured with a VAH, using a planar microphone array with 24 microphones, for one fixed orientation, in an anechoic and a reverberant room. Then, individual spectral weights for 185 orientations of the listener’s head were calculated with different parameter sets. Parameters included the number and the direction of the sources considered in the calculation of spectral weights as well as the required minimum mean white noise gain (WNGm). For both acoustical environments, the quality of the resulting synthesized BRIRs was assessed perceptually in head-tracked auralizations, in direct comparison to real loudspeaker playback in the room. Results showed that both rooms could be auralized with the VAH for speech signals in a perceptually convincing manner, by employing spectral weights calculated with 72 source directions from the horizontal plane. In addition, low resulting WNGmvalues should be avoided. Furthermore, in the dynamic binaural auralization with speech signals in this study, individual BRIRs seemed to offer no advantage over non-individual BRIRs, confirming previous results that were obtained with simulated BRIRs.

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.

A New Class of Differential Beamformers

Differential microphone arrays (DMAs) have been used in a wide range of applications for high-fidelity acoustic signal acquisition and enhancement. In the design of differential beamformers, three of the widely used measures are the directivity factor (DF), the front-to-back ratio (FBR), and the white noise gain (WNG). The former two have been used to obtain optimal differential beamformers, e.g., the hypercardioid and supercardioid, and the third one is generally used to analyze and control the robustness of the beamformer with respect to array imperfections due to sensors’ self noise, mismatch among sensors, and sensors’ placement errors. In this paper, we present a new measure called directivity factor and front-to-back ratio (DFBR), which is a generalization of DF and FBR. With this new measure, three different kinds of beamformers are derived. The first one is the maximum DFBR beamformer, which is deduced by maximizing DFBR with a joint diagonalization method. The second one is the $\psi$ -cardioid beamformer, which is the maximum DFBR beamformer corresponding to a distortionless constraint. The last one is the reduced-rank differential beamformer, which is obtained by properly choosing the dimension of the signal subspace and maximizing WNG subject to the distortionless constraint. The developed beamformers have many interesting properties, which are justified by both simulations and experiments.

On the Design of Sparse Arrays With Frequency-Invariant Beam Pattern

Beamformer performs spatial filtering to preserve the desired signal while suppressing interfering signals and noise arriving from directions other than the direction of interest. However, the beam pattern of the conventional beamformer is dependent on the frequency of the signal. It is common to use dense and uniform arrays for a broadband signal to achieve some essential performances together, such as frequency-invariant, white noise gain, directivity factor, front-to-back ratio, etc. Recently, the interest in sparse arrays is growing, mainly due to the capacity to reduce the number of sensors. Nevertheless, in general, finding a suitable sparse array layout is still a challenging task. Many studies have focused on optimization procedures to seek the sparse array deployment. This paper presents an alternative approach to determine the location of sensors. Starting with a weight spectrum of a virtual uniform array, some techniques are used, such as analyzing the weight spectrum to determine the critical sensors, applying the clustering technique to group the sensors into the different groups, and selecting the representative sensors for each group. After the sparse array deployment is specified, the optimization technique is applied to find the beamformer coefficients. The proposed method helps to save the computation time in the design phase, and its beamformer performance outperforms other state-of-the-art methods in several aspects.

Heterophasic Binaural Differential Beamforming for Speech Intelligibility Improvement

Differential beamformers with small-size microphone arrays are very attractive for audio and speech signal acquisition thanks to their high directivity and frequency-invariant spatial responses. However, such beamformers often suffer from significant white noise amplification at low frequencies, which makes their implementation in real-world systems challenging. One widely used way to circumvent this issue is to increase the number of microphones in the design of a given order differential beamformer, leading to the so-called robust differential beamformer in which the redundancy provided by the additional sensors are used to improve the white noise gain (WNG). But even with this robust solution, white noise amplification at low frequencies still exists. In this article, instead of trying to improve WNG, we adopt a method to render the white noise in such a way that it affects less the perception of the speech signal of interest. Specifically, we propose a binaural differential beamforming method in which a differential beamformer is designed with two sub-beamforming filters that simultaneously generates two outputs, one for the left ear and the other for the right ear. Motivated by psychoacoustic experiments, we design these two filters in such a way that they are orthogonal so that the coherence between the white noise at the binaural outputs is minimized while the coherence between diffuse noise is maximized. With the proposed binaural differential beamformers, the desired signal components and (amplified) white noise at the binaural differential beamformer's outputs are rendered into different directions or zones. Consequently, the human perception system can better distinguish the desired signal from white noise for improved speech intelligibility. The superiority of the proposed binaural beamforming technique is justified by simulations, experiments, as well as listening tests.

Random matrix theory analysis of the dominant mode rejection beamformer white noise gain

The dominant mode rejection (DMR) beamformer [Abraham and Owsley, IEEE Oceans (1990)] is an adaptive spatial processor that has been effectively used to null interference sources in scenarios where the number of training samples to estimate the sample covariance matrix (SCM) is limited. In these scenarios the ratio of the SCM dimension (array size) to the number of snapshots (training samples) is close to or even greater than one. Whereas classical asymptotic SCM results require the number of samples grow to infinity while keeping the dimension fixed, random matrix theory (RMT) provides a robust mathematical framework to analyze the SCM spectrum in snapshot-limited scenarios. RMT predictions of the SCM eigenvectors have already been applied to predict DMR beamformer notch depth in a single interferer environment (Buck and Wage, IEEE SSP (2012)]. This talk presents an RMT-based asymptotic approximation of the DMR SCM inverse in a loud multi-interferer environment. The RMT SCM model is used to predict the white noise gain (WNG) of the DMR beamformer, accounting for the impact when bulk spectral components are included in the dominant subspace. The performance of the RMT-based WNG prediction is compared to the sample mean using Monte Carlo simulations. [Funded by ONR 321US.]