Search

Linear differential microphone arrays (LDMAs) are commonly integrated into thin and portable devices to achieve high-fidelity speech acquisition. Traditional LDMAs typically consist of only omnidirectional microphones, which impose limitations on their ability to produce steerable spatial responses due to constraints in array element directivity and linear array geometry. A recent solution to this limitation involves integrating both omnidirectional and bidirectional microphones in LDMA design, enabling the creation of steerable spatial responses. This paper extends the core idea of integrating omnidirectional and bidirectional microphones, and develops a more general and comprehensive theory and method for designing steerable LDMAs. It makes two main contributions. Firstly, it introduces a general approach to designing steerable LDMAs, in which any type of directional microphones can be used. Secondly, it gives the minimum number of omnidirectional and directional microphones required to achieve a specific order of steerable LDMA. Simulations validate the proposed method and illustrate how omnidirectional and directional sensors can be combined to form the desired LDMAs.

Acoustic echo cancellation (AEC) is a crucial task in full duplex communications. As conventional linear filtering approaches are ineffective to deal with double-talk, various semi-blind source separation (SBSS)-based AEC algorithms are deceived, most of which are formulated and implemented in the frequency domain based on the multiplicative transfer function (MTF) model for computational efficiency. To avoid large latency and in order to deal with loudspeaker nonlinearities, the convolutive transfer function (CTF) model and odd power series expansion are leveraged, which are employed by numerous SBSS-based nonlinear AEC (SBSS-NAEC) algorithms. Conventional SBSS-NAEC methods estimate the series expansion coefficients and the CTF filter simultaneously making the number of free parameters to estimate large. Hence, the corresponding algorithms are computationally expensive and are difficult to optimize. In this work, we propose to decouple the series expansion coefficients and the CTF filters into a bilinear form and present a bilinear alternating optimization framework for estimating the model parameters. An alternating iterative projection (AIP) algorithm and an alternating element-wise iterative source steering (AEISS) algorithm are proposed. As the bilinear representation consists of less parameters compared to the conventional methods, the proposed algorithms not only improve the AEC performance but also reduce the computational complexity, which is validated by comprehensive simulations and experiments.

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

Personal sound zone (PSZ) refers to the technique that uses an array of loudspeakers and digital signal processing tools to achieve spatial soundfield control. To generate the target sound zones, this technique generally requires to know the acoustic transfer functions (ATFs) between the loudspeakers and the spots where soundfields are to be controlled. In practical applications, however, the true ATFs are never accessible and they have to be measured or estimated. Due to many sophisticated reasons, the measured ATFs generally deviate from the true ones, which may lead to significant degradation in performance of sound zone reproduction. In this work, a robust pressure matching (RPM) algorithm is presented for sound zone generation. It exploits a complex Gaussian mixture model (CGMM) to model the ATFs and their perturbations. The CGMM parameters are estimated using the expectation-maximization (EM) algorithm. To improve the robustness of the pressure matching method, an uncertainty constraint is applied to the ATF estimates and the pressure matching problem is then formulated as one of biconvex optimization. The coordinate descent algorithm is subsequently used to solve the optimization problem, thereby obtaining the optimal control filter. In comparison with the existing pressure matching methods without considering the effect of ATF perturbations, the presented algorithm is able to achieve lower normalized signal distortion energy and higher signal to interference ratio. Numerical simulations justify the effectiveness of the presented algorithm as well as its advantages over the traditional methods.

Differential microphone arrays (DMAs) have demonstrated a great potential for solving the high-fidelity sound acquisition problem in a wide range of applications as they possess many good properties such as frequency-independent beampatterns with high directivity. A significant number of efforts have been devoted to the design of DMAs and the associated beamformers. As a result, many different types of DMAs and differential beamforming methods have been developed over the last few decades, some of which have been successfully deployed in real systems and commercial products. However, given an application, how to design a DMA to achieve optimal performances is still an open issue. This work studies the problem of designing linear DMAs (LDMAs) from a geometric perspective. Based on the fundamental observation that most practical and interesting DMA beampatterns have nulls in some directions, we define a criterion based on the orthogonality between the beamforming filter and the steering vector in the nulls' directions. We then derive a family of differential beamformers by optimizing the defined criterion, some of which are well known but derived from a different perspective, while others are new. Simulations and experiments are carried out, and the results validate the proposed method and developed differential beamformers.

This paper studies the design of maximum directivity factor (MDF) beamformers based on uniform linear arrays (ULAs) consisting of acoustic vector sensors (AVSs). We first derive the main lobe constraints, which ensure that the beamformer's beampattern achieves a maximum in the look direction, and prove that any beamformer that satisfies the proposed constraints can be written as the sum of two orthogonal beamformers: the maximum white noise gain (MWNG) beamformer and a reduced-rank beamformer. Then, we derive the MDF beamformer by maximizing the directivity factor (DF) under the deduced constraints. We also derive a robust version of the MDF beamformer, which can keep the WNG above a pre-specified level. Compared to the conventional MDF beamformer based on ULAs with omnidirectional microphones, the designed MDF beamformer with uniform linear AVS arrays (ULAVSAs) can steer the beampattern to any look direction in the 3-dimensional space and achieves a higher directivity. The proposed MDF beamformer also outperforms the two-step MDF beamformer with ULAVSAs since it maximizes the DF. The proposed methods are validated through simulations as well as real experiments.

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). 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. 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.

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.

While differential beamforming with uniform linear arrays (ULAs) has been widely studied, there is little work so far regarding the design of differential beamformers with nonuniform linear arrays (NULAs). This article attempts to shed some light on the principles of differential beamforming with NULAs. We define spatial difference operators with NULAs, where any order of the spatial difference of the observation signals can be represented as the product of a nonuniform spatial difference operator matrix and the observation vector. Consequently, the design of differential beamformers is performed in two stages. In the first one, a nonuniform spatial difference operator matrix is applied to the array observations, thereby yielding differential signals. In the second stage, beamformers are designed and applied to the obtained differential signals to optimize the array performance. Based on the defined spatial difference operators, we derive from some performance metrics a family of differential beamformers with NULAs, which include the maximum directivity factor (DF), the maximum white noise gain (WNG), and the maximum front-to-back ratio (FBR) differential beamformers. To compromise between the DF and array robustness, we also derive the parameterized maximum DF and parameterized maximum FBR differential beamformers. The null-constraint maximum DF and WNG differential beamformers are also developed so that some nulls can be placed in specified directions for interference suppression. Simulation results validate the theoretical analysis and justify the properties of the proposed methods.

This paper studies the problem of microphone array beamforming to enhance a speech signal of interest in adverse acoustic environments, where interference and additive background noise coexist. The problem is formulated as one of convex optimization whose solution under a specified level of interference attenuation leads to an interference controlled maximum noise reduction (ICMR) beamformer, which can be expressed as a linear combination of two MVDR beamformers: one attempts to extract the desired source signal while the other attempts to extract the interference. The combination coefficients are functions of the array manifold vectors, noise coherence matrix, and the specified interference attenuation factor. By tuning the interference attenuation factor, the ICMR beamformer can be implemented to achieve aggressive interference attenuation or even eliminate interference completely; but this may lead to less additive noise suppression or even noise amplification. To control the maximum sacrifice in gain (SG) of the signal-to-noise ratio (SNR) that is acceptable for additive reduction, a variant of ICMR is derived, which is named as the ICMR-SG beamformer. Simulations are performed and the results show that the ICMR beamformer is able to control the amount of interference attenuation. In comparison, ICMR-SG controls the maximum SG of SNR while achieving the optimal possible level of interference attenuation.

End-to-end speaker verification achieves the verification through estimating directly the similarity score between a pair of utterances, which is formulated as a binary (i.e., target versus non-target) classification problem. Unlike the stage-wise method, an end-to-end verification approach optimizes the evaluation metrics directly and its output layer is parameter-free, which can save great computing and memory resources. However, it faces two important difficulties. The first one is how to deal with severely imbalanced trials, i.e., the number of target trials is much smaller than that of nontarget trials, and the other is about how to handle easy trials that do not help improve the model in training. To circumvent these two issues, we propose in this paper a binary cross-entropy (BCE) type of loss function and present a method to train the deep neural network (DNN) models based on the proposed loss function for end-to-end speaker verification. The training process employs a bipartite ranking method to deal with the trial imbalance problem and a curriculum learning method to help improve both the training stability and performance of the model by selecting non-target trials from easy to hard ones gradually along the convergence process. Since the training process employs bipartite ranking and curriculum learning and the loss function is of the generalized BCE form, we name the new approach curriculum bipartite ranking weighted binary cross-entropy (CBRW-BCE). Experimental results show that the model trained with CBRW-BCE not only achieves the state-of-the-art performance but is also well calibrated.

Reverberation, whichis caused by late reflections, impairs not only speech quality but also intelligibility. Consequently, dereverberation, a process to mitigate the impact of reverberation, has attracted significant research interests. Numerous approaches have been developed in the literature, among which the weighted-prediction-error (WPE) one has demonstrated promising potential for reducing or eliminating reverberation. The WPE method has been well studied and several variants have been developed. The adaptive one, called adaptive WPE (AWPE) method, has been widely investigated for use in real applications as it can deal with reverberation in time-varying acoustic environments. However, the computational complexity of AWPE is high, which may be a problem for its implementation in real-time systems. This paper presents some new insights into AWPE-based speech dereverberation by introducing the concepts of Kronecker product and partially time-varying filtering. It then develops two algorithms for dereverberation with lower complexity than AWPE. The significant contributions of this work are as follows. First, we propose a Kronecker product filtering framework for speech dereverberation, where the linear prediction filter is formulated as the Kronecker product of two sets of shorter filters. Second, we propose a partially time-varying Kronecker product filter for dereverberation. Instead of estimating the entire linear prediction filter as in the conventional method, the proposed one only needs to update part of the filter. The proposed approaches can significantly reduce the computational complexity without sacrificing dereverberation performance as compared to AWPE. Simulation results validate the theoretical analysis and justify the advantages of the new methods.

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.

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.

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.

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.

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.

Differential microphone arrays (DMAs) can achieve high directivity and frequency-invariant spatial response with small apertures; they also have a great potential to be used in a wide spectrum of applications for high-fidelity sound acquisition. Although many efforts have been made to address the design of linear DMAs (LDMAs), most developed methods so far only work for the situation where the source of interest is incident from the endfire direction. This paper studies the steering problem of differential beamformers with linear microphone arrays. We present new insights into beam steering of LDMAs and propose a series of steerable differential beamformers. The major contributions of this paper are as follows. 1) A series of ideal functions are defined to describe the ideal, target beampatterns of LDMAs. 2) We prove that first-order differential beamformers with linear microphone arrays are not steerable and their mainlobes can only be at the endfire directions. 3) We deduce the fundamental conditions for designing steerable differential beamformers with LDMAs. 4) We develop a method to design steerable beamformers with LDMAs using null constraints. Simulations and experiments validate the properties of the developed method.

Differential microphone arrays (DMAs) often encounter white noise amplification, especially at low frequencies. If the array geometry and the number of microphones are fixed, one can improve the white noise amplification problem by reducing the DMA order. With the existing differential beamforming methods, the DMA order can only be a positive integer number. Consequently, with a specified beampattern (or a kind of beampattern), reducing this order may easily lead to over compensation of the white noise gain (WNG) and too much reduction of the directivity factor (DF), which is not optimal. To deal with this problem, we present in this article a general approach to the design of DMAs with fractional orders. The major contributions of this article include but are not limited to: 1) we first define a directivity pattern that can achieve a continuous compromise between the pattern corresponding to the maximum DMA order and the omnidirectional pattern; 2) by approximating the beamformer's beampattern with the Jacobi-Anger expansion, we present a method to find the proper differential beamforming filter so that its beampattern matches closely the target directivity pattern of fractional orders; and 3) we show how to determine analytically the proper fractional order of the DMA with a given target beampattern when either the value of the DF or WNG is specified, which is useful in practice to achieve the desired beampattern and spatial gain while maintaining the robustness of the DMA system.

Receiver operating characteristic (ROC) and detection error tradeoff (DET) curves are two widely used evaluation metrics for speaker verification. They are equivalent since the latter can be obtained by transforming the former's true positive y-axis to false negative y-axis and then re-scaling both axes by a probit operator. Real-world speaker verification systems, however, usually work on part of the ROC curve instead of the entire ROC curve given an application. Therefore, we propose in this article to use the area under part of the ROC curve (pAUC) as a more efficient evaluation metric for speaker verification. A Mahalanobis distance metric learning based back-end is applied to optimize pAUC, where the Mahalanobis distance metric learning guarantees that the optimization objective of the back-end is a convex one so that the global optimum solution is achievable. To improve the performance of the state-of-the-art speaker verification systems by the proposed back-end, we further propose two feature preprocessing techniques based on length-normalization and probabilistic linear discriminant analysis respectively. We evaluate the proposed systems on the major languages of NIST SRE16 and the core tasks of SITW. Experimental results show that the proposed back-end outperforms the state-of-the-art speaker verification back-ends in terms of seven evaluation metrics.

This article presents a theoretical study of differential beamforming with uniform linear arrays. By defining a forward spatial difference operator, any order of the spatial difference of the observed signals can be represented as a product of a difference operator matrix and the microphone array observations. Consequently, differential beamforming is implemented in two stages, where the first one obtains spatial difference of the observations and the second stage optimizes the beamformer. The major contributions of this article are as follows. First, we propose a new theory of differential beamforming with uniform linear arrays, which shows clearly the connection between the conventional differential beamforming and the null-constrained differential beamforming methods. This provides some new insight into the design of differential beamformers. Second, we deduce some new differential beamformers, where conventional beamforming may be seen as a particular case. Specifically, we derive the maximum white noise gain (MWNG), maximum directivity factor (MDF), parameterized MDF, and parameterized maximum front-to-back ratio differential beamformers. Third, we further extend the idea of how to design optimal differential beamformers by combining both the observed signals and their spatial differences.

We study differential beamforming from a graph perspective. The microphone array used for differential beamforming is viewed as a graph, where its sensors correspond to the nodes, the number of microphones corresponds to the order of the graph, and linear spatial difference equations among microphones are related to graph edges. Specifically, for the first-order differential beamforming with an array of M microphones, each pair of adjacent microphones are directly connected, resulting in M - 1 spatial difference equations. On a graph, each of these equations corresponds to a 2-clique. For the second-order differential beamforming, each three adjacent microphones are directly connected, resulting in M - 2 second-order spatial difference equations, and each of these equations corresponds to a 3-clique. In an analogous manner, the differential microphone array for any order-of-differential beamforming can be viewed as a graph. From this perspective, we then derive a class of differential beamformers, including the maximum white noise gain beamformer, the maximum directivity factor one, and optimal compromising beamformers. Simulations are presented to demonstrate the performance of the derived differential beamformers.

Differential microphone arrays (DMAs) have many interesting properties and have been widely used in acoustic, audio, and speech applications. A critical part of a DMA is the differential beamformer, which is generally designed in two important steps: 1) specifying a target beampattern based on what differential sound pressure field the DMA is expected to respond to and 2) designing the differential beamforming filter so that the resulting beampattern matches the target one. Most efforts in the study of DMAs so far have focused on the second step while choosing one of the limited patterns available in the literature as the target beampattern. Since it governs how the array performs, how to design the target beampattern is an important problem, which this paper addresses. The major contributions of this paper consists of the following four aspects. First, a positive superposition theorem is presented, which shows that the linear combination of effective beampatterns with non-negative coefficients is always an effective beampattern. Second, we propose a general approach to the design of target DMA beampatterns based on the positive superposition theorem. Third, an overview of the classical target beampatterns is provided and discussion is made on how to form effective base patterns. Fourth, we show that the smallest first null of a DMA is π(2N) with N being the DMA order, which provides the rule of setting nulls in practice. Finally, with examples, we show that with the use of the alternating-direction-method-of-multipliers algorithm, the proposed approach is able to generate useful DMA target beampatterns.

This paper is devoted to beamforming with small-spacing microphone arrays for processing broadband and scattered acoustic sources. It presents a maximum diffuse noise gain (MDNG) beamformer in this context using the joint diagonalization technique, which is effective in suppressing diffuse and directional noise, but at a price of low white noise gain (WNG). We also introduce a maximum WNG (MWNG) beamformer, which is robust to the array imperfections, but paying a price of sacrificing the diffuse noise gain (DNG). To make a tradeoff between WNG and DNG so that the beamformer, on the one hand, can achieve high directivity and, on the other hand, is robust to implement, we propose a generalized MDNG beamformer, which includes both the MDNG and MWNG beamformers as particular cases. Simulations are conducted to illustrate the properties and advantages of the proposed beamformers.

Differential beamformers have attracted much interest over the past few decades. In this paper, we introduce differential Kronecker product beamformers that exploit the structure of the steering vector to perform beamforming differently from the well-known and studied conventional approach. We consider a class of microphone arrays that enable to decompose the steering vector as a Kronecker product of two steering vectors of smaller virtual arrays. In the proposed approach, instead of directly designing the differential beamformer, we break it down following the decomposition of the steering vector, and show how to derive differential beamformers using the Kronecker product formulation. As demonstrated, the Kronecker product decomposition facilitates further flexibility in the design of differential beamformers and in the tradeoff control between the directivity factor and the white noise gain.

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.

Blind multichannel identification is a challenging problem in many domains. The normalized multichannel frequency-domain least-mean-square (NMCFLMS) algorithm was developed to blindly identify a single-input multiple-output acoustic system, which can yield good performance in noise-free environments. However, the robustness of this algorithm to noise has been shown to be problematic. One way to improve the robustness is by applying a constraint on the spectral flatness of the channel impulse responses, which led to the development of the so-called robust normalized multichannel frequency-domain least-mean-square (RNMCFLMS) algorithm. This spectral flatness constraint, however, may not be always proper or reasonable in realistic acoustic environments. In this paper, we develop an ℓp-norm constraint based robust normalized multichannel frequency-domain least-mean-square (ℓp-RNMCFLMS) algorithm. The ℓp-norm constraint is introduced into the NMCFLMS algorithm to control the effect of different ℓp-norm penalties on the adaptive filter for the impulse responses with different degrees of sparseness. Numerical and realistic experiments justify the effectiveness of the proposed ℓp-RNMCFLMS algorithm.

This paper deals with two critical issues about uniform circular arrays (UCAs): frequency-invariant response and steering flexibility. It focuses on some optimal design of frequency-invariant beampatterns in any desired direction along the sensor plane. The major contributions are as follows. 1) We explain how to include the steering information in the desired directivity pattern. 2) We show that the optimal approximation of the beamformer's beampattern with a UCA from a least-squares error perspective is the Jacobi–Anger expansion. 3) We develop an approach to the design of any desired symmetric directivity pattern, where the deduced beampattern is almost frequency invariant and its main beam can be pointed to any wanted direction in the sensor plane. 4) With the proposed approach, we derive an explicit form of the white noise gain (WNG) and the directivity factor (DF), and explain clearly the white noise amplification problem at low frequencies and the DF degradation at high frequencies. The analysis also indicates that increasing the number of microphones can always improve the WNG. We show that the proposed method is a generalization of circular differential microphone arrays. The relationship between the proposed method and the so-called circular harmonics beamformers is also discussed.

Superdirective beamforming has attracted a significant amount of research interest in speech and audio applications, since it can maximize the directivity factor (DF) given an array geometry and, therefore, is efficient in dealing with signal acquisition in diffuse-like noise environments. However, this beamformer is very sensitive to sensor self-noise and mismatch among sensors, which considerably restricts its use in practical systems. This paper develops an approach to superdirective beamforming based on the Krylov matrix. We show that the columns of a proposed Krylov matrix, which span a chosen dimension of the whole space, are interesting beamformers; consequently, all different linear combinations of those columns lead to beamformers that have good properties. In particular, we develop the Krylov maximum white noise gain and Krylov maximum DF beamformers, which are obtained by maximizing the WNG and the DF, respectively. By properly choosing the dimension of the Krylov subspace, the developed beamformers that can make a compromise between reasonable values of the DF and white noise amplification. We also extend the basic idea to the design of the Krylov maximum front-to-back ratio, parametric superdirective, and parametric supercardioid beamformers.

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

Differential beamforming is one of the most popular beamforming approaches, which has the great potential to form frequency-invariant directivity patterns. In this paper, we study the design of beampatterns with multiple nulls in the same direction, which is clearly different from the design of beampatterns with distinct nulls. Our contributions are as follows. First, we show how to constrain multiple nulls to the same direction and design the desired beampattern with both the traditional and robust approaches. Second, we derive an explicit form of the white noise gain (WNG) of the traditional approach as a function of the frequency, interelement spacing, and null direction, which shows that the cardioid is the optimal beampattern as far as the WNG is concerned. Third, we prove that the WNG improvement of the robust approach rarely depends on the null direction at low frequencies. Finally, considering the fact that the robust differential beamforming approach may produce a frequency-dependent beampattern while improving the WNG, we develop a weighted-norm approach that can make a good compromise between the robustness of differential beamforming with respect to white noise and the frequency-invariant beampattern. The performance of the developed approach is verified by simulations.

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

Differential microphone arrays (DMAs), due to their small size and enhanced directivity, are quite promising in speech enhancement applications. However, it is well known that differential beamformers have the drawback of white noise amplification, which is a major issue in the processing of wideband signals such as speech. In this paper, we focus on the design of robust DMAs. Based on the Maclaurin’s series approximation and frequency-independent beampatterns, the robust first-, second-, and third-order DMAs are proposed by using more microphones than the order plus one, and the corresponding minimum-norm filters are derived. Compared to the traditional DMAs, the proposed designs are more robust with respect to white noise amplification while they are capable of achieving similar directional gains.

Linear microphone arrays combined with the minimum variance distortionless response (MVDR) beamformer have been widely studied in various applications to acquire desired signals and reduce the unwanted noise. Most of the existing array systems assume that the desired sources are in the broadside direction. In this paper, we study and analyze the performance of the MVDR beamformer as a function of the source incidence angle. Using the signal-to-noise ratio (SNR) and beampattern as the criteria, we investigate its performance in four different scenarios: spatially white noise, diffuse noise, diffuse-plus-white noise, and point-source-plus-white noise. The results demonstrate that the optimal performance of the MVDR beamformer occurs when the source is in the endfire directions for diffuse noise and point-source noise while its SNR gain does not depend on the signal incidence angle in spatially white noise. This indicates that most current systems may not fully exploit the potential of the MVDR beamformer. This analysis does not only help us better understand this algorithm, but also helps us design better array systems for practical applications.

The noise reduction problem is traditionally approached in the time, frequency, or transform domain. Having a signal dependent transform has shown some advantages over the traditional signal independent transform. Recently, the single-channel noise reduction problem in the Karhunen-Loève expansion (KLE) domain has received special attention. In this paper, the noise reduction problem in the KLE domain is studied from a multichannel perspective. We present a new formulation of the problem, in which inter-channel and inter-mode correlations are optimally exploited. We derive different optimal noise reduction filters and present a set of useful performance measures within this framework. The performance of the different filters is then evaluated through experiments in which not only noise but also competing speech sources are present. It is shown that the proposed multichannel formulation is more robust to competing speech sources than the single-channel approach and that a better compromise between noise reduction and speech distortion can be obtained.