News Column

Patent Issued for Method for Reconstructing an Acoustic Field

June 3, 2014



By a News Reporter-Staff News Editor at Information Technology Newsweekly -- A patent by the inventors Hald, Jorgen (Fredensborg, DK); Gomes, Jesper (Frederiksberg, DK), filed on June 26, 2009, was published online on May 20, 2014, according to news reporting originating from Alexandria, Virginia, by VerticalNews correspondents.

Patent number 8731851 is assigned to Bruel & Kjaer Sound & Vibration Measurement A/S (Naerum, DK).

The following quote was obtained by the news editors from the background information supplied by the inventors: "Near-field Acoustical Holography (NAH) is a very useful tool for 3D visualization of sound radiation and for precise noise source localization based on measurements over a surface near the sound source. Its ability to reconstruct also the evanescent wave components ensures a very high spatial resolution.

"A known Near-field Acoustical Holography method is based on regular-grid measurements across a level surface in a separable coordinate system, allowing the NAH calculation to be performed by spatial Discrete Fourier Transform (DFT), see e.g. E. G. Williams, J. D. Maynard, and E. J. Skudrzyk, 'Sound source reconstruction using a microphone array,' J. Acoust. Soc. Am. 68, 340-344 (1980). Due to the use of DFT, the processing is very fast, but a side effect of using the DFT includes severe spatial windowing effects unless the measurement area fully covers the areas with high sound pressure. In some cases this requirement on the measurement area cannot be fulfilled, and in many cases the necessary size becomes prohibitively large.

"A set of techniques have been proposed to reduce the spatial windowing effects, while still maintaining the DFT spatial processing but at the cost of an increased complexity and computational demands, see e.g. J. Hald, 'Reduction of spatial windowing effects in acoustical holography,' Proceedings of Inter-Noise 1994. Typically an iterative procedure is first used to extrapolate the measured sound pressure outside the measured area, followed by application of a DFT based holography method on the extended data window.

"Other methods have been proposed that seek to avoid the use of spatial DFT and to provide a reduction in the required measurement area.

"One such method is the Helmholtz' Equation Least Squares (HELS) method which uses a local model of the sound field in terms of spherical wave functions, see e.g. U.S. Pat. No. 6,615,143, Z. Wang and S. F. Wu, 'Helmholtz equation-least-squares method for reconstructing the acoustic pressure field,' J. Acoust. Soc. Am, 102(4), 2020-2032 (1997); or S. F. Wu, 'On reconstruction of acoustic fields using the Helmholtz equation-least-squares method,' J. Acoust. Soc. Am, 107, 2511-2522 (2000). However, since only spherical wave functions with a common origin are used to represent the sound field, errors will be introduced in the sound field reconstruction on the source surface, unless the source surface is also spherical and centered in the same origin. Another drawback of the above prior art method is that it requires a large number of measurement positions to obtain a sufficiently accurate model. A third drawback is that traditional regularization methods like Tikhonov regularization do not work properly. Instead, the above prior art method applies a computationally expensive iterative search for an optimal truncation of the spherical wave expansion combined with a least squares solution without regularization.

"Another previously proposed method is the Statistically Optimized Near-field Acoustic Holography (SONAH) method disclosed in R. Steiner and J. Hald, 'Near-field Acoustical Holography without the errors and limitations caused by the use of spatial DFT,' Intern. J. Acoust. Vib. 6, 83-89 (2001). However, this prior art method is based on plane wave functions and does not allow for accurate reconstruction of the sound field on non-planar source surfaces."

In addition to the background information obtained for this patent, VerticalNews journalists also obtained the inventors' summary information for this patent: "According to a first aspect, disclosed herein is a method of reconstructing a sound field based on a set of measurements, the method comprising: receiving measured values of a first acoustic quantity measured at a set of measurement locations; defining a set of virtual source locations and a set of wave functions, each wave function being representative of a respective sound field originating from a respective one of the defined set of virtual source locations; computing a second acoustic quantity at a target location from a superposition of the set of wave functions multiplied by respective expansion coefficients; wherein computing comprises determining the one or more expansion coefficients from a least-norm fit of the superposition of the set of wave functions to the received measured values.

"The inventors have realised that embodiments of the method and apparatus described herein allow an accurate reconstruction of a sound field even when the number of measurement points is smaller than the total number of wave functions used in the acoustic model. In particular, embodiments of the method described herein use the least-norm solution formula for determining the expansion coefficients of the acoustic model so as to handle underdetermined estimation problems, i.e. where the number of unknown expansion coefficients is larger than the number of measurements. Consequently, an efficient technique is provided that allows a detailed modelling even of sources with complicated surface geometries with a relatively small number of measurements and at relatively low computational cost.

"Generally, the term reconstructing a sound field refers to any process for predicting, e.g. by computation/estimation of approximate values one or more acoustic quantities, e.g., sound pressure, particle velocity, intensity and/or sound power, in a set of points, on the basis of measurements of the same or different acoustic quantity/quantities in a different (or the same) set of points.

"It will be appreciated that each virtual source location may be the origin for one or more wave functions, i.e. several wave functions may originate in one virtual source location.

"According to a second aspect, disclosed herein is a method of reconstructing a sound field at least one target location, the sound field being generated by at least one sound source, the method comprising: receiving one or more measured values of a first acoustic quantity measured at a set of measurement locations; defining a set of virtual source locations and a set of wave functions each representative of a respective sound field originating from a respective one of the defined set of virtual source locations; computing a second acoustic quantity for at least one target location from a superposition of the set of wave functions multiplied by respective expansion coefficients, the measurement locations and the at least one target location being located in a first source-free region, and the virtual source locations being located outside the first source-free region; wherein defining comprises defining a respective scaling surface for each of the defined virtual source locations, the scaling surface being located outside the first source-free region; and scaling each of the set of wave functions to have a predetermined amplitude on the scaling surface of the corresponding virtual source location.

"Hence, in embodiments of the method described herein the wave functions, e.g. multipole and/or spherical wave functions of different order around the respective virtual sound source locations, are scaled to have equal, or at least approximately equal, amplitude on scaling surfaces around their respective virtual sound sources. In particular, all wave functions having the same virtual source location as their origin may be scaled on the same scaling surface around that virtual source location. The scaling surfaces may be spherical surfaces having their centre in the corresponding virtual source location. In particular, when the scaling surfaces are spherical surfaces all having the same radius, the wave functions are scaled to have the same amplitude at a predetermined distance from their respective virtual source location (i.e. the wave function origins).

"Hence, embodiments of the method described herein apply a scaling of the wave functions in such a way that functions with stronger decay in the model region are scaled to lower amplitudes in the same region, in particular at the measurement locations. The scaling scheme described herein is particularly useful when the set of wave functions comprises multipole and/or spherical wave functions of different order, e.g. in an embodiment with a single virtual source location and a plurality of wave functions defined with the single virtual source locations as an origin. Since the wave functions have (approximately) equal amplitudes on the respective scaling surfaces, those wave functions with a stronger decay have reached a lower level when entering the first source-free region, in particular at the measurement locations, compared to those that have a weaker decay. Since these fast decaying waves have smaller amplitudes at the measurement locations, they will get a lower weight in the system of equations set up for calculation of the expansion coefficients, and they will therefore be the first to be cut away by a regularization process. This type of behaviour is desirable for the regularization methods to be able to retain field components well above the noise floor of the measured data and to cut away wave components that have decayed to so low levels that they are dominated by measurement inaccuracies. It is therefore an advantage of this scaling that it allows regularization schemes to perform optimal filtering, thereby allowing for a more accurate reconstruction.

"It will be appreciated that embodiments of the scaling described herein may be used both in connection with the HELS method and with the least-norm method described herein.

"The sound source may be any object emitting or reflecting acoustic radiation. The object may be arbitrarily shaped, and the sound may be any kind of sound, e.g. noise, audible sound, inaudible sound such as ultra-sound or infra-sound, etc., or a combination thereof.

"The first acoustic quantity at a measurement location may be measured by any suitable acoustic measuring device, e.g. a microphone, a hydrophone, a pressure gradient transducer, a particle velocity transducer, etc. or a combination thereof. In some embodiments, the measurement is performed by an array of acoustic measurement devices, e.g. a set of such devices arranged in a regular or irregular grid, for example, a two- or three-dimensional grid. The measured first acoustic quantity may be a sound pressure, a sound pressure gradient, a particle velocity, and/or the like.

"The reconstructed sound field may be represented by any suitable data set indicative of a spatial distribution of a second acoustic quantity such as a sound pressure, a sound intensity or a particle velocity. The first and second acoustic quantities may be the same quantity or different quantities. The data set may be represented as a map, e.g. a map of the second acoustic quantity directly on the actual surface geometry of an arbitrary surface, e.g. a surface of a sound/noise emitting object to be analysed or a surface close to the object.

"Since particle velocity on top of a hard surface corresponds closely to the actual vibration of the surface itself, the results can directly be used for correlation with structural models. In some embodiments the sound field is represented by a conformal map. In some embodiments, the sound field parameters are reconstructed and mapped on a surface, typically near the source surface. Alternatively or additionally, for example spectra at single positions in the 3D region of validity of the wave function representation may be reconstructed.

"A least-norm fit of the one or more scaled wave functions to the received measured values generally refers to a numerical process of determining a set of complex expansion coefficients that solve a set of linear equations where the determined set of factors has an at least approximately minimal norm, i.e. in a situation where the set of equations does not have a unique solution, the process determines a solution with a minimal norm among the set of possible solutions. The set of linear equations requires, at the measurement locations, the sum of wave functions, each multiplied with a respective expansion coefficient, to be equal the measured values. The expansion coefficients are complex numbers representing an amplitude and a phase. It has been realised by the inventors that an accurate reconstruction of the sound field may be achieved even when the solution to the set of equations is not unique, and the process determines a least-norm solution.

"It is a further advantage of embodiments of the method described herein that the computation of the second acoustic quantity from a least-norm solution formula may be based on the computation of correlation functions indicative of respective correlations of the set of wave functions at a first one of said measurement locations with the set of wave functions at a second location, the second location being chosen from the target location and the measurement locations. In particular, the least-norm solution formula involves a square correlation matrix having a dimensionality equal to the number of measurement locations, wherein each element of the square correlation matrix is indicative of a correlation of the set of wave functions evaluated at respective measurement locations.

"The measurement locations and the at least one target location may be located in a first source-free, homogeneous region, and the virtual source locations may be located outside the first source-free region, typically in a predetermined distance from the boundary of the first source free region. Each virtual source location serves as an origin of one or more of the set of wave functions.

"In some embodiments, the first source-free region is defined by a first surface, the at least one sound source is comprised in an object having an object surface, and at least a part of the object surface defines at least a part of the first surface. Hence the object surface may at least partly define the first source-free measurement and reconstruction domain in which the sound field may be reconstructed, thereby allowing at least some target positions to be located on the object surface. The virtual source locations are defined outside the measurement and reconstruction domain.

"For example, the virtual source locations, i.e. the origins of respective ones of the wave functions, may be defined at a predetermined spatial relation to the object surface around the sound source to be analysed. For example, the virtual sources may be defined at locations all having a predetermined distance from the object surface, thereby providing an accurate mapping of the surface. Generally, the object surface may have a first side facing the sound source and the virtual source locations and a second side facing the measurement and target locations. However, in some embodiments some virtual source locations may be located outside the source object, but outside the first source-free region.

"The wave functions may be a suitable set of elementary wave functions, e.g. an orthogonal set of basis functions. The elementary wave functions each satisfy the so-called Helmholtz equation or reduced wave equation in the reconstruction region. In some embodiments, the elementary wave functions are spherical wave functions. In some embodiments, for each virtual source location a limited set of scaled spherical wave functions is used, e.g. to define a finite multipole expansion, i.e. a linear combination of a finite number of multipole basis functions, e.g. chosen from a monopole point function, a set of dipole point functions (with different orientations), and/or the like. In particular, the process may compute the second acoustic quantity directly or indirectly from a linear combination of the elementary wave functions, each elementary wave function being weighted by a respective expansion coefficient. For example, the computation may comprise computing the spatial derivatives of the wave functions. If the linear combination of wave functions provides the pressure, then the particle velocity can be obtained from the weighted sum of spatial derivatives of the wave functions (using the same expansion coefficients, but with an additional frequency dependent constant factor). Also, the sound intensity may be calculated from the pressure and particle velocity.

"It is noted that features of the methods described above and in the following may be implemented at least in part in software or firmware and carried out on a data processing device or other processing means caused by the execution of program code means such as computer-executable instructions. Here and in the following, the term processing means comprises any circuit and/or device suitably adapted to perform the above functions. In particular, the above term comprises general- or special-purpose programmable microprocessors, Digital Signal Processors (DSP), Application Specific Integrated Circuits (ASIC), Programmable Logic Arrays (PLA), Field Programmable Gate Arrays (FPGA), special purpose electronic circuits, etc., or a combination thereof.

"Embodiments of the present invention can be implemented in different ways, including the methods described above and in the following, systems, devices and product means, each yielding one or more of the benefits and advantages described in connection with one of the first-mentioned methods, and each having one or more embodiments corresponding to the embodiments described in connection with one of the first-mentioned methods and/or as disclosed in the dependent claims.

"In particular, embodiments of a processing device for reconstructing a sound field comprise an interface for receiving measured values of a first acoustic quantity measured at a set of measurement locations, and a processing unit.

"A system for reconstructing a sound field may comprise an apparatus as disclosed above and in the following and a set of transducers for measuring the first acoustic quantity at a set of measurement locations, and connectable in communication connection to the apparatus so as to forward the measured values to the apparatus.

"A computer program may comprise program code means adapted to cause a data processing system to perform the steps of the method disclosed above and in the following when the program code means are executed on the data processing system. The computer program may be stored on a storage means or embodied as a data signal. The storage means may comprise any suitable circuitry or device for storing data, such as a RAM, a ROM, an EPROM, EEPROM, flash memory, magnetic or optical storage device, such as a CD ROM, a DVD, a hard disk, and/or the like."

URL and more information on this patent, see: Hald, Jorgen; Gomes, Jesper. Method for Reconstructing an Acoustic Field. U.S. Patent Number 8731851, filed June 26, 2009, and published online on May 20, 2014. Patent URL: http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&p=33&u=%2Fnetahtml%2FPTO%2Fsearch-bool.html&r=1640&f=G&l=50&co1=AND&d=PTXT&s1=20140520.PD.&OS=ISD/20140520&RS=ISD/20140520

Keywords for this news article include: Information Technology, Information and Data Processing, Bruel & Kjaer Sound & Vibration Measurement A/S.

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Source: Information Technology Newsweekly