Uncertainty analysis of the standard delay-and-sum beamformer and array calibration

Beamforming has become an ubiquitous task in aeroacoustic noise measurements for source localization and power estimation. The standard delay-and-sum (DAS) beamformer is the most commonly used beamforming algorithm due to its simplicity and robustness and also serves as the basis for more sophisticated algorithms, such as the deconvolution approach for the mapping of acoustic sources (DAMAS). The DAS data reduction equation is a function of many parameters including the microphone locations, microphone transfer functions, temperature and the cross-spectral matrix (CSM), where each one of these parameters has a unique uncertainty associated with it. This paper provides a systematic uncertainty analysis of the DAS beamformer and Dougherty's widely used calibration procedure under the assumption that the underlying mathematical model of incoherent, monopole sources is correct. An analytical multivariate method based on a first-order Taylor series expansion and a numerical Monte-Carlo method based on assumed uncertainty distributions for the input variables are considered. The uncertainty of calibration is analyzed using the Monte-Carlo method, whereas the uncertainty of the DAS beamformer is analyzed using both the complex multivariate and the Monte-Carlo methods. It is shown that the multivariate uncertainty analysis method fails when the perturbations are relatively large and/or the output distribution is non-Gaussian, and therefore the Monte-Carlo analysis should be used in the general case. The calibration procedure is shown to greatly reduce the uncertainties in the DAS power estimates. In particular, 95 percent confidence intervals for the DAS power estimates are presented with simulated data for various scenarios. Moreover, the 95 percent confidence intervals for the integrated DAS levels at different frequencies are computed using experimental data. It is shown that with experimental data, the 95 percent confidence intervals for the integrated power levels are within of the mean levels when the component uncertainties are set at low but achievable values.     

Optimum array design to maximize Fisher information for bearing estimation

Source bearing estimation is a common application of linear sensor arrays. The Cramer-Rao bound (CRB) sets a lower bound on the achievable mean square error (MSE) of any unbiased bearing estimate. In the spatially white noise case, the CRB is minimized by placing half of the sensors at each end of the array. However, many realistic ocean environments have a mixture of both white noise and spatially correlated noise. In shallow water environments, the correlated ambient noise can be modeled as cylindrically isotropic. This research designs a fixed aperture linear array to maximize the bearing Fisher information (FI) under these noise conditions. The FI is the inverse of the CRB, so maximizing the FI minimizes the CRB. The elements of the optimum array are located closer to the array ends than uniform spacing, but are not as extreme as in the white noise case. The optimum array results from a trade off between maximizing the array bearing sensitivity and minimizing output noise power variation over the bearing. Depending on the source bearing, the resulting improvement in MSE performance of the optimized array over a uniform array is equivalent to a gain of 2-5 dB in input signal-to-noise ratio.     

Numerical simulation of tonal fan noise of computers and air conditioning systems

Current approaches to fan noise simulation are mainly based on the Lighthill equation and socalled aeroacoustic analogy, which are also based on the transformed Lighthill equation, such as the wellknown FW-H equation or the Kirchhoff theorem. A disadvantage of such methods leading to significant modeling errors is associated with incorrect solution of the decomposition problem, i.e., separation of acoustic and vortex (pseudosound) modes in the area of the oscillation source. In this paper, we propose a method for tonal noise simulation based on the mesh solution of the Helmholtz equation for the Fourier transform of pressure perturbation with boundary conditions in the form of the complex impedance. A noise source is placed on the surface surrounding each fan rotor. The acoustic fan power is determined by the acoustic-vortex method, which ensures more accurate decomposition and determination of the pressure pulsation amplitudes in the near field of the fan.     

Inverse problem with beamforming regularization matrix applied to sound source localization in closed wind-tunnel using microphone array

Microphone arrays have become a standard technique to localize and quantify source in aeroacoustics. The simplest approach is the beamforming that provides noise source maps with large main lobe and strong side lobes at low frequency. Since a decade, the focus is set on deconvolution techniques such as DAMAS or Clean-SC. While the source map is clearly improved, these methods require a large computation time. In this paper, we propose a sound source localization technique based on an inverse problem with beamforming regularization matrix called Hybrid Method. With synthetic data, we show that the side lobes are removed and the main lobe is narrower. Moreover, if the sound noise source map provided by this method is used as input in the DAMAS process, the number of DAMAS iterations is highly reduced. The Hybrid Method is applied to experimental data obtained in a closed wind-tunnel. In both cases of acoustic or aeroacoustic data, the source is correctly detected. The proposed Hybrid Method is found simple to implement and the computation time is low if the number of scan points is reasonable.     

A covariance fitting approach for correlated acoustic source mapping

Microphone arrays are commonly used for noise source localization and power estimation in aeroacoustic measurements. The delay-and-sum (DAS) beamformer, which is the most widely used beamforming algorithm in practice, suffers from low resolution and high sidelobe level problems. Therefore, deconvolution approaches, such as the deconvolution approach for the mapping of acoustic sources (DAMAS), are often used for extracting the actual source powers from the contaminated DAS results. However, most deconvolution approaches assume that the sources are uncorrelated. Although deconvolution algorithms that can deal with correlated sources, such as DAMAS for correlated sources, do exist, these algorithms are computationally impractical even for small scanning grid sizes. This paper presents a covariance fitting approach for the mapping of acoustic correlated sources (MACS), which can work with uncorrelated, partially correlated or even coherent sources with a reasonably low computational complexity. MACS minimizes a quadratic cost function in a cyclic manner by making use of convex optimization and sparsity, and is guaranteed to converge at least locally. Simulations and experimental data acquired at the University of Florida Aeroacoustic Flow Facility with a 63-element logarithmic spiral microphone array in the absence of flow are used to demonstrate the performance of MACS.     

A fast signal subspace approach for the determination of absolute levels from phased microphone array measurements

Phased microphone arrays are used in a variety of applications for the estimation of acoustic source location and spectra. The popular conventional delay-and-sum beamforming methods used with such arrays suffer from inaccurate estimations of absolute source levels and in some cases also from low resolution. Deconvolution approaches such as DAMAS have better performance, but require high computational effort. A fast beamforming method is proposed that can be used in conjunction with a phased microphone array in applications with focus on the correct quantitative estimation of acoustic source spectra. This method bases on an eigenvalue decomposition of the cross spectral matrix of microphone signals and uses the eigenvalues from the signal subspace to estimate absolute source levels. The theoretical basis of the method is discussed together with an assessment of the quality of the estimation. Experimental tests using a loudspeaker setup and an airfoil trailing edge noise setup in an aeroacoustic wind tunnel show that the proposed method is robust and leads to reliable quantitative results.     

Passive noise control by enhancing aeroacoustic interference due to structural discontinuities in close proximity

In-duct devices are commonly installed in flow ducts for various flow management purposes. The structural construction of these devices indispensably creates disruption to smooth flow through duct passages so they exist as structural discontinuities in duct flow. The presence of these discontinuities provides additional possibility of noise generation. In real practice, in-duct devices do not exist alone in any duct system. Even though each in-duct device would generate its own noise, it might be possible that these devices could be properly arranged so as to strengthen the interference between individual noise; thus giving rise to an overall reduction of noise radiation in the in-duct far field. This concept of passive noise control is investigated by considering different configurations of two structural discontinuities of simple form (i.e., a cavity) in tandem in an unconfined flow and in opposing setting within a flow duct. It is known that noise generated by a cavity in unconfined domain (unconfined cavity) is strongly dependent on flow-resonant behavior within the cavity so the interference it produces is merely aeroacoustic. The objective of the present study is to verify the concept of passive noise reduction through enhancement of aeroacoustic interference due to two cavities by considering laminar flow only. A two-dimensional approach is adopted for the direct aeroacoustic calculations using a direct numerical simulation (DNS) technique. The position and geometries of the cavities and the Mach number are varied; the resultant aeroacoustic behavior and acoustic power are calculated. The numerical results are compared with a single cavity case to highlight the effect of introducing additional cavities to the aeroacoustic problem. Resonant flow oscillations occur when two unconfined cavities are very close and the associated acoustic field is very intense with no noise reduction possible. However, for duct aeroacoustics, it is found that a 7.9 db reduction of acoustic power in the downstream side of the duct or a total reduction of ∼6 db is possible with opposing cavities having an offset of half a cavity length. In addition, the reduction is shown to be free from lock-on with trapped modes of the ducts with cavities.     

Error bounds in diffusion tensor estimation using multiple-coil acquisition systems

We extend the diffusion tensor (DT) signal model for multiple-coil acquisition systems. Considering the sum-of-squares reconstruction method, we compute the Cramér–Rao bound (CRB) assuming the widely accepted noncentral chi distribution. Within this framework, we assess the effect of noise in DT estimation and other measures derived from it, as a function of the number of acquisition coils, as well as other system parameters. We show the applications of CRB in many actual problems related to DT estimation: we compare different gradient field setup schemes proposed in the literature and show how the CRB can be used to choose a convenient one; we show that for fiber-type anisotropy tensors the ellipsoidal area ratio (EAR) can be estimated with less error than other scalar factors such as the fractional anisotropy (FA) or the relative anisotropy (RA), and that for this type of anisotropy tensors, increasing the number of coils is equivalent to increasing the signal-to-noise ratio, i.e., the information of the different coils can be regarded as independent. Also, we present results showing the CRB of several parameters for actual DT-MRI data. We conclude that the CRB is a valuable tool to optimal experiment design in DT-related studies.     

Limits of coherence-based aeroacoustic analysis in the presence of distributed sources

Coherence-based analysis techniques utilizing a small number of microphones are often applied in aeroacoustic measurements. These techniques can remove statistically incoherent noise, electronic or hydrodynamic, from acoustic signals measured by microphones, at significantly lower cost than array methods. However, the assumptions involved in the usage of the ordinary coherence function technically limit analysis to a single-source field. In the presence of multiple sources the coherence function breaks down and ordinary analysis techniques under-predict true acoustic levels. This phenomenon is demonstrated mathematically and illustrated using experimental trailing edge noise data.     

Parametric time domain analysis of the multiple input/scalar output problem: The source identification problem

A multiple input/scalar output stationary time series identification problem is considered from a parametric model time domain point of view. Particular emphasis is on the source identification problem. Closed form formula estimates of the individual source power contributions are expressed in terms of sample correlations that are obtained from the observed input and output time series and from parametric models fitted to that data. The estimates of the noise power contributions are asymptotically jointly normally distributed. The mean values and covariance matrix of those estimates yield confidence interval estimates of the individual and joint power contributions.The motivation for developing a rational polynomial transfer function or ARMA model of the multi-input scalar output plus additive noise situation is given. A two correlated input/single output version of this model is considered for a Monte Carlo simulation study. Parametric ARMA and approximate AR models are fitted to the simulated data. The asymptotic normality, and the distribution of the mean and covariances of the source power contribution computed from the ARMA and AR models are appraised.Several facets of the relative performance of windowed periodogram and AR model spectral analysis are examined for the multiple input/scalar output identification problem. The points that are emphasized are that conventional windowed periodogram spectral analysis is subjective, not particularly satisfactory for the sharp spectral peak situation that is commonly encountered in vibration data analysis and very likely not as good as “objective” Akaike criterion order AR modelled spectral analysis.     

Sparsity constrained deconvolution approaches for acoustic source mapping

Using microphone arrays for estimating source locations and strengths has become common practice in aeroacoustic applications. The classical delay-and-sum approach suffers from low resolution and high sidelobes and the resulting beamforming maps are difficult to interpret. The deconvolution approach for the mapping of acoustic sources (DAMAS) deconvolution algorithm recovers the actual source levels from the contaminated delay-and-sum results by defining an inverse problem that can be represented as a linear system of equations. In this paper, the deconvolution problem is carried onto the sparse signal representation area and a sparsity constrained deconvolution approach (SC-DAMAS) is presented for solving the DAMAS inverse problem. A sparsity preserving covariance matrix fitting approach (CMF) is also presented to overcome the drawbacks of the DAMAS inverse problem. The proposed algorithms are convex optimization problems. Our simulations show that CMF and SC-DAMAS outperform DAMAS and as the noise in the measurements increases, CMF works better than both DAMAS and SC-DAMAS. It is observed that the proposed algorithms converge faster than DAMAS. A modification to SC-DAMAS is also provided which makes it significantly faster than DAMAS and CMF. For the correlated source case, the CMF-C algorithm is proposed and compared with DAMAS-C. Improvements in performance are obtained similar to the uncorrelated case.     

Flap-edge aeroacoustic measurements and predictions

An aeroacoustic model test has been conducted to investigate the mechanisms of sound generation on high-lift wing configurations. This paper presents an analysis of flap side-edge noise, which is often the most dominant source. A model of a main element wing section with a half-span flap was tested at low speeds of up to a Mach number of 0.17, corresponding to a wing chord Reynolds number of approximately 1.7 million. Results are presented for flat (or blunt), flanged, and round flap-edge geometries, with and without boundary-layer tripping, deployed at both moderate and high flap angles. The acoustic database is obtained from a small aperture directional array (SADA) of microphones, which was constructed to electronically steer to different regions of the model and to obtain farfield noise spectra and directivity from these regions. The basic flap-edge aerodynamics is established by static surface pressure data, as well as by computational fluid dynamics (CFD) calculations and simplified edge flow analyses. Distributions of unsteady pressure sensors over the flap allow the noise source regions to be defined and quantified via cross-spectral diagnostics using the SADA output. It is found that shear layer instability and related pressure scatter is the primary noise mechanism. For the flat edge flap, two noise prediction methods based on unsteady-surface-pressure measurements are evaluated and compared to measured noise. One is a new causality spectral approach developed here. The other is a new application of an edge-noise scatter prediction method. The good comparisons for both approaches suggest that the prediction models capture much of the physics. Areas of disagreement appear to reveal when the assumed edge noise mechanism does not fully define the noise production. For the different edge conditions, extensive spectra and directivity are presented. The complexity of the directivity results demonstrate the strong role of edge source geometry and frequency in the noise radiation. Significantly, for each edge configuration, the spectra for different flow speeds, flap angles, and surface roughness were successfully scaled by utilizing aerodynamic performance and boundary-layer scaling methods developed herein.     

Estimation of the detection range of a hydroacoustic system based on the acoustic power flux receiver

Approaches to estimating the detection range of systems based on vector receivers are considered. The approaches rely on a detailed analysis of the process of signal’s acoustic power flux formation in the presence of ambient sea noise and uncover the signal information parameters at the receiver output that provide the required statistically confident range of weak signal detection under these conditions. Based on the sonar equations and the known fundamental relationships between the outputs of a pressure receiver and a vector receiver for signal and noise, estimates of the maximum possible gain in the detection range of an acoustic power flux receiver are considered as a function of anisotropy of the ambient noise field in the area.     

An experimental study of airfoil instability tonal noise with trailing edge serrations

This paper presents an experimental study of the effect of trailing edge serrations on airfoil instability noise. Detailed aeroacoustic measurements are presented of the noise radiated by an NACA-0012 airfoil with trailing edge serrations in a low to moderate speed flow under acoustical free field conditions. The existence of a separated boundary layer near the trailing edge of the airfoil at an angle of attack of 4.2 degree has been experimentally identified by a surface mounted hot-film arrays technique. Hot-wire results have shown that the saw-tooth surface can trigger a bypass transition and prevent the boundary layer from becoming separated. Without the separated boundary layer to act as an amplifier for the incoming Tollmien–Schlichting waves, the intensity and spectral characteristic of the radiated tonal noise can be affected depending upon the serration geometry. Particle Imaging Velocimetry (PIV) measurements of the airfoil wakes for a straight and serrated trailing edge are also reported in this paper. These measurements show that localized normal-component velocity fluctuations that are present in a small region of the wake from the laminar airfoil become weakened once serrations are introduced. Owing to the above unique characteristics of the serrated trailing edges, we are able to further investigate the mechanisms of airfoil instability tonal noise with special emphasis on the assessment of the wake and non-wake based aeroacoustic feedback models. It has been shown that the instability tonal noise generated at an angle of attack below approximately one degree could involve several complex mechanisms. On the other hand, the non-wake based aeroacoustic feedback mechanism alone is sufficient to predict all discrete tone frequencies accurately when the airfoil is at a moderate angle of attack.     

平面叶栅气动噪声的高精度谱元方法求解

本文建立了一种用于气动噪声预测的高精度三角谱元方法。结合CBS法(Characteristic-based split method)求解Navier-Stokes方程获取伪声压声源后,基于波动方程求解声传播问题。谱元法的谱收敛特性满足了气动声学问题的高精度需求,而CBS法的引入保证了高雷诺数问题的计算稳定性。通过两组基准解问题求解验证了本文方法的正确性。将本文方法应用于平面叶栅气动噪声计算,并考察了不同攻角下噪声的变化规律。研究内容为进一步探究各类流体机械的气动噪声提供了一种新的途径。     

Helicopter noise in hover: Computational modelling and experimental validation

The aeroacoustic characteristics of a helicopter rotor are calculated by a new method, to assess its applicability in assessing rotor performance in hovering. Direct solution of the Euler equations in a noninertial coordinate system is used to calculate the near-field flow around the spinning rotor. The far-field noise field is calculated by the Ffowcs Williams–Hawkings (FW–H) method using permeable control surfaces that include the blade. For a multiblade rotor, the signal obtained is duplicated and shifted in phase for each successive blade. By that means, the spectral characteristics of the far-field noise may be obtained. To determine the integral aerodynamic characteristics of the rotor, software is written to calculate the thrust and torque characteristics from the near-field flow solution. The results of numerical simulation are compared with experimental acoustic and aerodynamic data for a large-scale model of a helicopter main rotor in an open test facility. Two- and four-blade configurations of the rotor are considered, in different hover conditions. The proposed method satisfactorily predicts the aerodynamic characteristics of the blades in such conditions and gives good estimates for the first harmonics of the noise. That permits the practical use of the proposed method, not only for hovering but also for forward flight.     

Independent component analysis in the presence of noise in fMRI

A noisy version of independent component analysis (noisy ICA) is applied to simulated and real functional magnetic resonance imaging (fMRI) data. The noise covariance is explicitly modeled by an autoregressive (AR) model of order 1. The unmixing matrix of the data is determined using a variant of the FastICA algorithm based on Gaussian moments. The sources are estimated using the principle of maximum likelihood by modeling the source densities as asymmetric exponential functions. Effect of dimensionality reduction on the effective noise covariance used, accuracy of the obtained mixing matrix and degree of improvement in estimating fMRI sources are investigated. The primary conclusions after using this method of evaluation are as follows: (a) weighting matrix estimates are similar for noisy and conventional ICA in the realm of typical fMRI data, and (b) source estimates are improved by 5% (as measured by the correlation coefficient) in realistic simulated data by explicitly modeling the source densities and the noise, even when just a simple white noise model is used.     

各向同性噪声场中单矢量传感器虚源消除MUSIC测向方法

针对各向同性噪声场中,单矢量水声传感器声压通道和振速通道的噪声功率不一致性造成MUSIC测向方法性能急剧下降的问题,提出了一种增秩MUSIC测向方法和一种幅度加权MUSIC测向方法。增秩MUSIC测向方法通过增加信号子空间维度消除虚源的影响,在单目标情况和低信噪比时可获得高精度方位估计结果,其性能接近克拉美洛界限(CRB),但它不能分辨双目标;幅度加权MUSIC测向方法通过对接收信号幅度加权消除了虚源,并且能分辨双目标。      

矢量气动声学的理论研究进展及应用

气动声学的声比拟理论以密度、声压等标量为波动算子变量,建立非齐次波动方程,描述流体运动及与边界作用诱发声音的辐射,但标量无法直接描述声能量的传播过程和途径.在流体力学研究中,标量用于描述当前当地的物质状态,而矢量用于描述质量和能量的传输.借鉴上述思想,开展了矢量气动声学的研究,概述矢量气动声学的理论研究进展及应用,主要包括:（1）以声粒子速度为变量,采用声比拟理论的思想直接从Navier-Stokes方程出发推导建立了气动声学的矢量波动方程及两种频域解;（2）综合利用声压和声粒子速度的积分解,直接求解声源周围的瞬时和有功声强矢量场,直观显示声能量的传播途径,应用于旋转声源辐射声能量的传播分析,揭示了亚音速旋转声源辐射声能量的3种传播模式:螺旋模式、声学黑洞模式和R-A模式;（3）采用球谐级数展开方法建立旋转点/紧凑声源辐射噪声的声压和声粒子速度的频域解析解,在此基础上推导了声功率谱的频域解析解,建立了识别旋转叶片声源在空间域和频域分布特征的方法;（4）综合利用矢量气动声学方法和等效源方法,显示声源和散射边界周围声强矢量场的分布特征和能量传播途径,直接揭示了阻抗边界主要的吸声位置以及直接计算得到阻抗边界的吸收声功率.      

Method of improved scatterer size estimation and application to parametric imaging using ultrasound

The frequency dependence of RF signals backscattered from random media (tissues) has been used to describe the microstructure of the media. The frequency dependence of the backscattered RF signal is seen in the power spectrum. Estimates of scatterer properties (average scatterer size) from an interrogated medium are made by minimizing the average squared deviation (MASD) between the measured power spectrum and a theoretical power spectrum over an analysis bandwidth. Estimates of the scatterer properties become increasingly inaccurate as the average signal to noise ratio (SNR) over the analysis bandwidth becomes smaller. Some frequency components in the analysis bandwidth of the measured power spectrum will have smaller SNR than other frequency components. The accuracy of estimates can be improved by weighting the frequency components that have the smallest SNR less than the frequencies with the largest SNR in the MASD. A weighting function is devised that minimizes the noise effects on the estimates of the average scatterer sizes. Simulations and phantom experiments are conducted that show the weighting function gives improved estimates in an attenuating medium. The weighting function is applied to parametric images using scatterer size estimates of a rat that had developed a spontaneous mammary tumor.