Instrumentation for Noise Measurement and Analysis
3.15 SOUND SOURCE LOCALISATION
3.15.1 Nearfield Acoustic Holography (NAH)
Acoustic holography involves the measurement of the amplitude and phase of a sound field at many locations on a plane at some distance from a sound source, but in its near field so that evanescent waves contribute significantly to the microphone signals. The measurements are than used to predict the complex acoustic pressure and particle velocity on a plane that approximates the surface of the source (the prediction plane).
Multiplying these two quantities together gives the sound intensity as a function of location in the prediction plane, which allows direct identification of the relative strength of the acoustic radiation from various areas of a vibrating structure. The theoretical analysis that underpins this technique is described in detail in a book devoted just to this topic (Williams, 1999) and so it will only be summarised here.
Acoustic holography may involve only sound pressure measurements in the measurement plane (using phase matched microphones) and these are used to predict both the acoustic pressure and particle velocity in the plane of interest. The product of these two predicted quantities is then used to determine the acoustic intensity on the
plane of interest which is usually adjacent to the sound radiating structure.
Alternatively, only acoustic particle velocity may be measured in the measurement plane (using the “Microflown”) and then these measurements may be used to predict the acoustic pressure as well as the acoustic particle velocity in the measurement plane. Finally, both acoustic pressure and acoustic particle velocity may be measured in the measurement plane. Then the acoustic particle velocity measurement is used to predict the acoustic particle velocity in the plane of interest and the acoustic pressure measurement is used to predict the acoustic pressure in the plane of interest.
Jacobsen and Liu (2005) showed that measurement of both acoustic pressure and particle velocity gave the best results followed by measurement of only particle velocity with the measurement of only acoustic pressure being a poor third. The reason that using like quantities to predict like quantities gives the best results is intuitively obvious, but the reasons that the particle velocity measurement gives much better results than the pressure measurement need some explanation. The reasons are summarised below:
$ Particle velocity decays more rapidly than sound pressure towards the edges of the measurement region and has a larger dynamic range. This means that spatial windowing, (similar to the time domain windowing discussed in Appendix D, but in this case in the spatial domain), which is a necessary part of the application of the technique to finite size structures, does not have such an influence when particle velocity measurements are used.
$ Predicting particle velocity from acoustic pressure measurements results in amplification of high spatial frequencies which increases the inherent numerical instability of the prediction. On the other hand, predicting acoustic pressure from acoustic particle velocity results in reduction of the amplitudes of high spatial frequencies.
$ Phase mismatch between transducers in the measurement array has a much greater effect on the prediction of acoustic particle velocity from acoustic pressure measurements than vice versa.
The practical implementation of acoustic holography is complex and expensive, involving many sensors whose relative phase calibration must be accurately known, and whose relative positions must be known to a high level of accuracy. However, commercially available instrumentation exists, which does most of the analysis transparently to the user, so it is relatively straightforward to use.
3.15.1.1 Summary of the Underlying Theory
Planar near field acoustic holography involves the measurement of the amplitudes and relative phases of either or both of the acoustic pressure and acoustic particle velocity at a large number of locations on a plane located in the nearfield of the radiating structure on which it is desired to locate and quantify noise sources. The measurement plane must be in the near field of the structure and as close as practical to it, but no closer than the microphone spacing. As the measurement surface is a plane, the noise source map will be projected on to a plane that best approximates the structural
K(kx,ky) ' m
4
&4m
4
&4
p(x,y,zm)ej(kxx%kyy)dxdy (3.49)
p(x,y,zm) ' 1 (2π)2m
4
&4m
4
&4
K(kx,ky)e&j(kxx%kyy)dkxdky (3.50)
kx2%ky2 # k2 (3.51)
kx2%ky2%kz2 ' k2 ' (ω/c)2 (3.52) surface. If the structural surface is curved, it is possible to use a curved sensor array but the analysis becomes much more complicated. The resolution (or accuracy of noise source location) of this technique is equal to the spacing of the measurement array from the noise radiating surface at low frequencies and at frequencies above the frequency where this distance is equal to half a wavelength, the resolution is half a wavelength. The following analysis is for a measurement array of infinite extent with microphones sufficiently close together that the measured sound pressure on the measurement plane is effectively continuous. The effect of deviating from this ideal situation by using a finite size array and a finite number of sensors will be discussed following the analysis.
For the analysis, it is assumed that the plane encompassing the radiating structure is in the x-y plane at a z coordinate, zp # 0 and the measurement plane is located at zm > 0 and is parallel to the plane at zp. The wavenumber transform corresponding to complex pressure measurements made on the measurement plane is a transform from the spatial domain to the wavenumber domain and may be written as:
where kx and ky are the wavenumber components in the x and y directions and their values cover the wavenumber range from - 4 to + 4. Of course, in practice the finite size of the array and the physical spacing of the microphones will limit the wavenumber spectrum and hence the frequency range that can be covered. The acoustic pressure is given by the inverse transform as:
From the preceding two equations, it can be seen that the wave number spectrum, K(kx,ky) for any given value of kx and ky may be interpreted as the amplitude of a plane wave provided that:
The plane wave propagates in the (kx, ky, kz) direction and kz must satisfy:
If Equation (3.51) is not satisfied, then the particular component of the wavenumber spectrum represents an evanescent wave whose amplitude decays with distance from the sound source.
The wavenumber transform in the prediction plane, z = zp can be calculated by multiplying Equation (3.50) with an exponential propagator defined as:
kz '
k2&kx2&ky2 for kx2%ky2#k2
&j kx2%ky2&k2 for kx2%ky2>k2
(3.55)
G(zp,zm,kx,ky) ' e&j kz(zp&zm) (3.53)
Gpu(zp,zm,kx,ky) ' kz
ρcke&j kz(zp&zm) (3.54)
Gup(zp,zm,kx,ky) ' ρck
kz e&j kz(zp&zm) (3.56)
After the multiplication of the wavenumber transform by G in Equation (3.53), the inverse transform (see Equation (3.55)) is taken to get the sound pressure in the prediction plane.
The acoustic particle velocity in the prediction plane is calculated from the wavenumber transform of Equation (3.49) by multiplying it by the pu propagator defined below and then taking the inverse transform, as for the pressure.
where:
It is also possible to complete the entire process using particle velocity measurements instead of pressure measurements. In this case, the particle velocity is substituted for the acoustic pressure in Equation (3.49) to give the particle velocity wave number transform and then Equation (3.50) gives the particle velocity on the LHS instead of the acoustic pressure. The propagator in equation (3.53) is then multiplied with the particle velocity wavenumber transform and then the inverse transform is taken as in Equation (3.50) to give the particle velocity on the prediction plane. The acoustic pressure on the prediction plane can be calculated by multiplying the particle velocity wavenumber transform by the following predictor prior to taking the inverse transform.
Implementation of the above procedure in practice requires the simplifications of a finite size measurement plane (which must be slightly larger than the radiating structure being analysed) and a finite spacing between measurement transducers. The adverse effect of the finite size measurement plane is minimised by multiplying the wavenumber transform by a spatial window that tapers towards the edge of the array so less and less weighting is placed on the measurements as one moves from the centre to the edge of the array. To prevent wrap around errors, the array used in the wavenumber transform is larger than the measurement array and all points outside the actual measurement array are set equal to zero.
The effect of the finite spacing between sensors in the measurement array is to limit the ability of the array to sample high spatial frequency components that exist if the sound field varies strongly as a function of location. To avoid these high frequency
Φk(x,y,z) ' e&j(kxx%kyy%kz(zp&zm)) (3.57)
p(x,y,z) . j
N n'1
anΦkn(x,y,z) (3.58)
components, the array must be removed some distance from the noise radiating structure but not so far that the evanescent modes are so low in intensity that they cannot be measured.
The other effect of finite microphone spacing is to limit the upper frequency for the measurement as a result of the spatial sampling resolution. In theory this spacing should be less than half a wavelength but in practice, good results are obtained for spacings less than about one quarter of a wavelength.
The lower frequency ability of the measurement is limited by the array size, which should usually be at least a wavelength, but in some special cases, if the sound pressure at the edges of the array has dropped off sufficiently, an array size of 1/3 of a wavelength can be used.
To be able to calculate the finite Fast Fourier Transforms, the measurement grid for NAH must be uniform; that is, all sensors must be uniformly spaced.
3.15.2 Statistically Optimised Nearfield Acoustic Holography (SONAH) The SONAH method (Hald et al., 2007) is a form of nearfield acoustic holography in which the FFT calculation is replaced with a least squares matrix inversion. The advantage of this method is that the measurement array does not need to be as large as the measurement source and the measurement sensors need not be regularly spaced.
In addition, the sound field can be calculated on a surface that matches the contour of the noise radiating surface. However, the computer power needed for the inversion of large matrices can be quite large. The noise source location resolution is similar to that for NAH.
The requirements for maximum microphone spacing are similar to NAH and typical maximum operating frequencies of commercially available SONAH systems range from 1 kHz to 6 kHz and the typical dynamic range (difference in level between strongest and weakest sound sources that can be detected) is 15 to 20 dB. There seems to be no lower limiting frequency specified by equipment manufacturers.
The array size affects the lower frequency limit of the measurement but the requirements for SONAH are much less stringent than those for NAH. It is possible to undertake measurements down to frequencies for which the array size is 1/8 of a wavelength.
The analysis begins by representing the sound field as a set of plane evanescent wave functions defined as:
where kz is defined by Equation (3.55). In practice, the sound field at any location, (x, y, z) is represented by N plane wave functions, chosen to cover the wave number spectrum of interest, so that:
p m '
p(x1,y1,z1) p(x2,y2,z2)
! p(xL,yL,zL)
and a ' a1 a2
! aN
and φ(x,y,z) ' Φk
1(x,y,z) Φk
2(x,y,z)
! Φk
N(x,y,z)
(3.62a,b,c) p(xR,yR,zR) . j
N n'1
anΦkn(xR,yR,zR) R ' 1,...,L (3.59)
p(x,y,z) ' a Tφ(x,y,z) (3.63)
a ' A HmA m%αI &1A Hm p m (3.64)
A m '
Φk1(x1,y1,z1) Φk2(x1,y1,z1) ... ΦkN(x1,y1,z1) Φk1(x2,y2,z2) Φk2(x2,y2,z2) ... ΦkN(x2,y2,z2)
! !
Φk1(xL,yL,zL) Φk2(xL,yL,zL) ... ΦkN(xL,yL,zL)
(3.60)
p m ' A m a (3.61)
α ' 1% 1
2(kd)2 × 10SNR/ 10 (3.65)
The coefficients, an are determined by using the pressure measurements over L locations, (xR, yR, zR, R = 1,...., L ), in the measurement array so that:
The plane wave functions corresponding to all the measurement points, L, can be expressed in matrix form as:
Then the pressure at the measurement locations may be written as:
where:
Equation (3.58) may then be used with Equation (3.62) to write an expression for the pressure at any other point not in the measurement array (and usually on the surface of the noise source being examined) as:
The matrix a is found by inverting Equation (3.61) which includes the measured data.
As the matrix is non-square a pseudo inverse is obtained so we obtain:
where I is the identity matrix and α is the regularisation parameter, which is usually selected with the following equation to give a value close to the optimum:
Φk(x,y,z) ' kz
ρcke&j(kxx%kyy%kz(zp&zm)) (3.66)
where SNR is the signal-to-noise ratio for the measured data and d = zm - zp is the distance between the sound source and the measurement plane.
The particle velocity is obtained using the same procedure and same equations as above, except that Equation (3.57) is replaced with:
This measurement technique is accurate in terms of quantifying the sound intensity as a function of location on the noise emitting structure and it has the same resolution and frequency range as NAH. No particle velocity measurement is needed.