Chao Pan

Performance Study of the MVDR Beamformer as a Function of the Source Incidence Angle

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.

On the design and implementation of linear differential microphone arrays

Differential microphone array (DMA), a particular kind of sensor array that is responsive to the differential sound pressure field, has a broad range of applications in sound recording, noise reduction, signal separation, dereverberation, etc. Traditionally, an Nth-order DMA is formed by combining, in a linear manner, the outputs of a number of DMAs up to (including) the order of N - 1. This method, though simple and easy to implement, suffers from a number of drawbacks and practical limitations. This paper presents an approach to the design of linear DMAs. The proposed technique first transforms the microphone array signals into the short-time Fourier transform (STFT) domain and then converts the DMA beamforming design to simple linear systems to solve. It is shown that this approach is much more flexible as compared to the traditional methods in the design of different directivity patterns. Methods are also presented to deal with the white noise amplification problem that is considered to be the biggest hurdle for DMAs, particularly higher-order implementations.

Theoretical analysis of differential microphone array beamforming and an improved solution

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 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.

On the noisereduction performance of the MVDR beamformer innoisy and reverberant environments

The minimum variance distortionless response (MVDR) beam-former has been widely studied for extraction of desired speech signals in noisy acoustic environments. The performance of this beam-former, however, depends on many factors such as the array geometry, the source incidence angle, the noise field characteristics, the reverberation conditions, etc. In this paper, we study the performance of the MVDR beamformer in different noise and reverberation conditions with a linear microphone array. Using the gain in signal-to-noise ratio (SNR) as the performance metric, we show that the optimal performance of the MVDR beamformer generally occurs when the source is in the endfire directions in different types of noise, which indicates that, as long as a linear array is used, we should configure it in such a way that the endfire direction is pointed to the desired source. Simulations in reverberant environments also verified this result, though the performance difference between end-fire and broadside directions reduces as the degree of reverberation increases.

Design of robust differential microphone arrays with orthogonal polynomials

Differential microphone arrays have the potential to be widely deployed in hands-free communication systems thanks to their frequency-invariant beampatterns, high directivity factors, and small apertures. Traditionally, they are designed and implemented in a multistage way with uniform linear geometries. This paper presents an approach to the design of differential microphone arrays with orthogonal polynomials, more specifically with Jacobi polynomials. It first shows how to express the beampatterns as a function of orthogonal polynomials. Then several differential beamformers are derived and their performance depends on the parameters of the Jacobi polynomials. Simulations show the great flexibility of the proposed method in terms of designing any order differential microphone arrays with different beampatterns and controlling white noise gain.

Design of directivity patterns with a unique null of maximum multiplicity

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.

Fundamentals of Differential Beamforming

This book provides a systematic study of the fundamental theory and methods of beamforming with differential microphone arrays (DMAs), or differential beamforming in short. It begins with a brief overview of differential beamforming and some popularly used DMA beampatterns such as the dipole, cardioid, hypercardioid, and supercardioid, before providing essential background knowledge on orthogonal functions and orthogonal polynomials, which form the basis of differential beamforming. From a physical perspective, a DMA of a given order is defined as an array that measures the differential acoustic pressure field of that order; such an array has a beampattern in the form of a polynomial whose degree is equal to the DMA order. Therefore, the fundamental and core problem of differential beamforming boils down to the design of beampatterns with orthogonal polynomials. But certain constraints also have to be considered so that the resulting beamformer does not seriously amplify the sensors self noise and the mismatches among sensors. Accordingly, the book subsequently revisits several performance criteria, which can be used to evaluate the performance of the derived differential beamformers. Next, differential beamforming is placed in a framework of optimization and linear system solving, and it is shown how different beampatterns can be designed with the help of this optimization framework. The book then presents several approaches to the design of differential beamformers with the maximum DMA order, with the control of the white noise gain, and with the control of both the frequency invariance of the beampattern and the white noise gain. Lastly, it elucidates a joint optimization method that can be used to derive differential beamformers that not only deliver nearly frequency-invariant beampatterns, but are also robust to sensors self noise.

Performance Measures Revisited

In this chapter, we revisit some of the most fundamental performance measures associated with differential beamforming.

Reduced-order robust superdirective beamforming with uniform linear microphone arrays

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.

Study and design of robust differential beamformers with linear microphone arrays

Differential beamformers can generate frequency-invariant spatial responses and therefore have the great potential to solve many broadband acoustic signal processing problems such as noise reduction, signal separation, dereverberation, etc. The design of such beamformers, however, is not a trivial task. This paper is devoted to the study and design of differential beamformers with linear array geometry. The objective is to design robust differential beamformers that can form frequency-invariant beampatterns. The major contribution consists of the following aspects. (1) It discusses a general approach to the design of linear DMAs that can use any number of microphones to design a given order DMA as long as the number of microphones is at least one more than the order of the DMA. (2) It presents a method that can maximize the white noise gain with a given number of microphones and order of the DMA; so the resulting beamformer is more robust to sensor noise than the beamformer designed with the traditional DMA method. (3) It discusses how to use nonuniform geometries to further improve robustness of differential beamformers. (4) It investigates the possibility to improve the robustness of differential beamformers with the use of the diagonal loading technique.