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Reference-less acoustic holography techniques based on sound intensity
Journal of Sound and Vibration 333 (2014) 3598–3608
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Reference-less acoustic holography techniques based on sound intensity A. Nejade n Institut National de Recherche et de Sécurité (INRS), 54519 Vandoeuvre Cedex, France
a r t i c l e in f o
abstract
Article history: Received 29 October 2013 Received in revised form 13 March 2014 Accepted 24 March 2014 Handling Editor: I. Lopez Arteaga Available online 29 April 2014
This paper describes a detailed implementation and application of the intensity-based holography method called BAHIM, and it investigates its possibilities and its limitations. Although this method is attractive in the sense that it bypasses the reference signal requirement, i.e., one of the main difficulties of the classical near-field acoustic holography, its degree of accuracy does not appear to be adequate. This constraint has consequently led to the development of another method called CIBNAH, based on complex intensity, as an alternative. This novel approach provides additional accuracy in detecting and localizing sources and good compatibility, in form and amplitude, with NAH results. & 2014 Elsevier Ltd. All rights reserved.
1. Introduction The application of classical near-field acoustic holography (NAH) to a structure requires a reference signal, when the sound field is generated by coherent sources. This application becomes more complex, however, when the sources are not or are partially coherent in which case they must each be represented by an ad hoc reference signal. This is a serious difficulty since in many cases, e.g., in machines, the sources are not necessarily identified: the identification being itself the preliminary aim of the holography. Numerous investigations have been carried out on systems that include partially coherent sources. The aim has been to decompose the total sound field into as many subfields as the sources. This task has been undertaken through several techniques such as ‘partial coherence method’ [1] or ‘virtual source method’ [2]. Hallman and Bolton remark that partial coherence decomposition would allow the partial fields to be associated with the fields radiated by individual sub-sources only if they are sufficiently separated spatially. This separation is not, however, described quantitatively. Application of the partial coherence technique requires a preliminary knowledge of the sources. ‘Virtual coherence and principle component analysis’ on the other hand is not particularly suitable to localize the sources [2]. This latter point is confirmed by Kim and Bolton [3]. Kwon et al. [4] present a comparison between singular value (introduced by Hald [5]) and partial coherence decomposition techniques and show that the reference signals must be captured at locations close to the individual real sources. This last condition is not always easy to satisfy (e.g., when source locations are unknown or inaccessible). To overcome such difficulties, some authors suggest the virtual reference approach [6] that in most cases consists of an iterative procedure where each step would result in more optimally placed and less numerous sources than the previous one. It is difficult to determine the number of such iterations, however, when the number and locations of the sources are unknown. Moreover, ending up with a number of virtual sources that exceeds that of the real sources could result in an
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http://dx.doi.org/10.1016/j.jsv.2014.03.031 0022-460X/& 2014 Elsevier Ltd. All rights reserved.
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inclusion of undesired noise source(s), as input(s), in the model. Kim and Bolton [3] undertake a more refined approach than that by Nam and Kim [6]. Here a number of virtual references are defined and then reduced by eliminating one between each pair of totally coherent sources. The optimized virtual sources are then treated through singular value decomposition (SVD) or partial coherence techniques to obtain partial fields associated with each. This technique again confirms the necessity that the virtual source locations be very close to the real sources, an exigency that may put the advantage of virtual reference technique into question. The above rather long, yet non-exhaustive, literature survey aimed at showing the complexity of implementation of the acoustic holography to the real physical systems, e.g., industrial machines: a persisting complexity, despite the numerous developments in the field. It is therefore desirable to search for alternative solutions that bypass the requirement for resorting to reference signals. During 1980s, Loyau and Pascal developed a method called broad band acoustic holography reconstruction from acoustic intensity measurements (BAHIM) [7] that could overcome the above complication since it did not need any reference signal. This method was accompanied by only a simple analytical example and was, later on, followed by a case history [8] without any theoretical or practical detail on the application. Although this method is fast and simple for practical applications, to the author's knowledge, it has not been followed by any further investigation. This paper presents an extended investigation of the BAHIM method of implementation, application on a structure as well as its limitations. It will be followed by the description of a newly developed broad band holography method called complex intensity based near-field acoustic holography (CIBNAH) that has proved to be more accurate than BAHIM, despite its theoretical simplicity. The common point between CIBNAH and BAHIM is the search for the phase of a complex entity (pressure or, in case of CIBNAH, pressure and/or particle velocity) on the hologram. They are, however, different from the theoretical and experimental points of view. Therefore, CIBNAH cannot be considered as an improved version, but as an alternative to BAHIM. 2. Theoretical background of BAHIM 2.1. Basic derivations ! The basic relation linking the pressure p to the active intensity vector I is given by 2 ! jpj ! γ; (1) I ¼ 2ρc ! ∇w ^ ∂w^ ∂w ^ ^ ^ ^ where ! γ ¼ k ¼ 1k ∂w ∂x i þ ∂y j þ ∂z k is the field factor, i; j et k are the unit vectors in x, y and z directions, respectively, ρ is the air density, c is the speed of sound, k is the acoustic wavenumber and w is the phase. Considering the components of the active intensity on the hologram plane, i.e., along the X and Y axes, relation (1) can be expressed as jpðx; yÞj2 ∂cðx; yÞ ^ ∂cðx; yÞ^ iþ j (2) I x ðx; yÞ^i þI y ðx; yÞ^j ¼ ∂x ∂y 2ρck The above relation can be written as i ∂cðx; yÞ^ ∂cðx; yÞ^ 2ρck h iþ j¼ I ðx; yÞ^i þ I y ðx; yÞ^j 2 x ∂x ∂y jpðx; yÞj
(3)
The implementation of relation (3), presented in space domain, is difficult. It is more practical when expressed in wavenumber kx et ky domain (corresponding to x and y directions) through spatial Fourier transformation: jkx ψðkx ; ky Þ^i jky ψðkx ; ky Þ^j ¼ k½ℱ fλx ðkx ; ky Þg^i þℱfλy ðkx ; ky Þg^j where ℱ represents the Fourier transform and 8 < λx ðkx ; ky Þ ¼ ℱ f2ρcI x ðx; yÞ=jpðx; yÞj2 g : λy ðkx ; ky Þ ¼ ℱ f2ρcIy ðx; yÞ=pðx; yÞ2 g
(4)
(5)
Using (4) and (5), the phase transform ψðkx ; ky Þ can be obtained as 2
2
2
2
ψðkx ; ky Þ ¼ jk½kx =ðkx þky Þλx ðkx ; ky Þ jk½ky =ðkx þky Þλy ðkx ; ky Þ
(6)
The phase in the spatial domain is then provided by the inverse spatial Fourier transform: ψ h ðx; yÞ ¼ ℱ 1 fψðkx ; ky Þg
(7)
Combination of this phase and the measured pressure amplitude then yields the complex pressure on the hologram: pffiffiffiffiffiffiffiffi where j ¼ 1:
ph ðx; yÞ ¼ jph ðx; yÞjejch ðx;yÞ
(8)
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2.2. Back propagation calculations Having obtained the complex pressures measured at all the grid points on the hologram located at a distance h from the surface of the studied structure, the usual holography procedures are applied using the following classical holography relation that calculates an entity γ (pressure, particle velocity) on the surface of the structure (xs ; ys ; zs ), from an entity χ measured on the hologram of coordinates (x; y; zh ) [9]: Z Z 1 γðxs ; ys ; zs Þ ¼ χðx; y; zh Þg p ðxs x; ys y; zs zh Þdx dy (9) 1
where gp represents the direct (pressure-pressure, velocity-velocity) or cross (pressure-velocity, velocity-pressure) green propagator. In most cases, it is easier to treat the above convolution in wavenumber domain, using Fourier transformation of (9) Γðkx ; ky ; zs Þ ¼ X ðkx ; ky ; zh Þ Gðkx ; ky ; zs zh Þ
(10)
where the terms Γ, X and G are the spatial double Fourier transforms of γ, χ and g, respectively. The expression for the propagator G depends on the measured entity on the hologram and the entity looked for through back propagation. In the direct holography (velocity back propagation U-U, pressure back propagation P-P), it is expressed as GUU ¼ GPP ¼ e jkz h ; (11) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 where h¼ zh zs and kz ¼ k kx ky . In case of “cross-holography”, the propagator takes other forms. For U-P, i.e., pressure calculation from velocity hologram measurements, used further in the article, the associated propagator has the form GUP ¼
ρck jkz h e : kz
(12)
To sum up, the application of BAHIM is carried out in two stages: (I) hologram complex pressure computation based on Eqs. (1)–(8) and (II) back propagation according to the classical NAH formulations expressed by relations (9) and (10). At the first stage of the computation (complex pressure), where a transformation from space to wavenumber domain is carried out, the usual classical holography procedures such as space windowing, zero-padding and k-space filtering do not seem necessary since all these techniques will be employed in the second stage, anyway. 2.3. Calculation process 2
2
2
2
The terms kx =ðkx þ ky Þ and ky =ðkx þ ky Þ in (6) are dependent on the interaction between the x and y components of the wavenumbers, i.e., kx ¼ 2πðnx =Lx Þ et ky ¼ 2πðny =Ly Þ where nx et ny are the wavenumber indices, represented in vector shape as: nx ¼[0, 1, 2, 3, …, ((mx =2) 1), 0, ((mx =2) 1), ((mx =2) 2), …, 3, 2, 1] and ny ¼[0, 1, 2, 3, …, ((my =2) 1), 0, ((my =2) 1), (my =2) 2), …, 3, 2, 1]0 , where mx et my are the measurement point numbers along kx and ky axes, respectively, and the sign 0 implies the transpose of the vector. One can observe that for certain values of ðnx , ny Þ, i.e. (1, 1), (1, mx/2), (my/2, 1) and (mx/2, my/2), the wavenumbers kx and ky simultaneously vanish, leading to 4 singularity points in (6). An example of such a situation is presented in Table 1, showing an 8 8 wavenumber matrix where singularities are designated by “undefined”. As mentioned earlier, these singularities occur at kx ¼ky ¼0, pointing to a plane wave that propagates uniquely in the z direction (i.e., perpendicular to the hologram). It means that the in-plane phase gradients are equal to zero. Physically speaking, this implies a DC offset, in the phase, whose value is undefined. We are, therefore, in the presence of an ill-posed problem since it does not satisfy the uniqueness of solution [10]. This represents the main computational difficulty in the application of BAHIM. As in all ill-conditioned problems, one must search for approximate solutions. We may replace the singularities either by the average value of the adjacent cells or by zeros following the behaviour of the associated columns. In both cases the singularity peaks disappear allowing all the other measurement points on the hologram to influence the
Table 1 An example of the matrix jkK x =ðK 2x þ K 2y Þ. Undefined 0.000E þ00 0.000E þ00 0.000E þ00 Undefined 0.000E þ00 0.000E þ00 0.000E þ00
0 0.7463i 0 0.3731i 0 0.1492i 0 0.0746i 0 0.7463i 0 0.0746i 0 0.1492i 0 0.3731i
0 0.3731i 0 0.2985i 0 0.1865i 0 0.1148i 0 0.3731i 0 0.1148i 0 0.1865i 0 0.2985i
0 0.2487i 0 0.2239i 0 0.1722i 0 0.1243i 0 0.2487i 0 0.1243i 0 0.1722i 0 0.2239i
Undefined 0.000E þ00 0.000E þ00 0.000E þ00 Undefined 0.000E þ00 0.000E þ00 0.000E þ00
0 þ0.2487i 0 þ0.2239i 0 þ0.1722i 0 þ0.1243i 0 þ0.2487i 0 þ0.1243i 0 þ0.1722i 0 þ0.2239i
0 þ0.3731i 0 þ0.2985i 0 þ0.1865i 0 þ0.1148i 0 þ0.3731i 0 þ0.1148i 0 þ0.1865i 0 þ0.2985i
0 þ0.7463i 0 þ0.37317i 0 þ0.1492i 0 þ0.0746i 0 þ0.7463i 0 þ0.0746i 0 þ0.1492 0 þ0.3731i
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final result in proportion to their amplitude. The consequence of the back propagation computation will be the same for both types of approximations provided that Npm⪢4, where Npm is the total number of the measurement points and 4 refers 2 2 to the number of singularity points in the jkkx =ðkx þ ky Þ matrix. 3. Complex intensity based near-field acoustic holography (CIBNAH), a novel method of reference-less holography, as alternative to BAHIM 3.1. Introduction BAHIM method was developed as an alternative allowing the bypass of the main difficulty of classical (or multireference) holography, namely, the choice of reference signal(s). To do so, the theoretical derivations aim at calculating the pressure phase distribution on the hologram, the information that, otherwise, can only be provided by a reference signal. However, as it will be demonstrated in Section 5, this method shows inconsistencies in its accuracy mainly because of the slow decay of the in-plane (x, y) components of the intensity, from which the phase distribution on the hologram is deduced. 3.2. CIBNAH theoretical background Holography with no need for reference signal(s) can otherwise be achieved in a simpler manner, by using the normal complex intensity obtained by the following basic and direct relation: ! !n I z ¼ 12 p v z
(13) ! n where v z represents normal particle velocity vector and its complex conjugate. CIBNAH consists of using the phase of this complex intensity along with the measured particle velocity (or pressure) module in spectral domain. This phase has essentially the same nature as the one calculated by BAHIM. Pressure–velocity probes easily available nowadays in the market allow simultaneous measurements of the two quantities (at the same point in space, up to 12 kHz) and consequently their relative phase. As mentioned in some works in the literature [11], a particle velocity field generated by a source undergoes a much faster decay in its plane than the pressure field. This leads to a better source detection and separation in the back propagation. The same observation was made during our investigations. For this reason, the focus, in the present work, has been on velocity–velocity or velocity–pressure holography, i.e., particle velocity back propagation using particle velocity hologram measurements. From (13), the complex particle velocity can be expressed in frequency domain as v~ z ¼
pffiffiffiffiffiffiffiffiffiffi Gpvz pffiffiffiffiffiffiffiffiffiffi ¼ Gvz vz ejch Gvz vz Gpvz
(14)
where G's represent the auto and cross-spectra corresponding to the entities p (pressure) and vz (normal particle velocity), both acquired simultaneously on the measurement plane (hologram) using a pressure–velocity probe. Sign represents a pffiffiffiffiffiffiffiffiffiffi complex value. The two terms Gpvz and Gvz vz , in the above formula, are therefore the outcomes of simultaneous spectral averages. The phase ch between the pressure and the particle velocity includes information both on spatial coordinates of the corresponding measurement point (on the hologram) and the eventual source signal fluctuations of the investigated structure. An alternative representation for (14) can be given by qffiffiffiffiffiffiffiffi Gpvz qffiffiffiffiffiffiffiffi v~ z ¼ Gpp ¼ H pvz Gpp ; (15) Gpp where the transfer function H pvz may provide a more consistent (and stable) phase for the complex velocity. As mentioned earlier, the lack of a need for reference signal(s) constitutes the basic advantage of CIBNAH over NAH, and especially the multi-reference NAH. There are also other potential gains in applying CIBNAH, namely,
Considerably lower number of the terms involved in the CIBNAH formulation leading to less possible computation errors. Inaccuracies due to approximate (or inappropriate) choice of reference signals required by NAH are absent. The experimental and computational processes are easier and faster. 4. Multi-reference NAH In several cases below, the results of CIBNAH will be compared to those of double reference near-field acoustic holography (NAH), using partial coherence techniques. This latter approach was undertaken since the sources were partially coherent. The data for both techniques (CIBNAH and NAH) were acquired simultaneously. For this reason, and because we are investigating strictly the same structure, the results from the two methods must be analogous. This comparison aims at verifying the accuracy of CIBNAH and also to show its insensitivity to the value of the coherence between the sources.
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5. BAHIM test setup A special structure was designed to test the method in a controlled manner. It is a 38 23 28 cm3 double chamber wooden box, each chamber half-full of absorbing materials and containing a source (loudspeaker) at its mid height. Circular apertures have been drilled on the top plate: one communicating with the smaller chamber and four with the larger one (Fig. 1a, b). This configuration enables us to vary certain testing parameters such as the distance between the sources while knowing their exact position (by obstructing three out of four apertures on the larger chamber). Results presented in this paper correspond to cases where the sound was emitted through the apertures 1 and 5 (14 cm source spacing) and through 3 and 5 (8 cm spacing). Also, through the choice of the input signals, coherence between the sources could be varied. Three sets of tests were performed with measurement surface (hologram) parallel to and above the box top plate at distances (h) of 25 mm, 55 mm (and 80 mm for BAHIM, uniquely). The top plate's dimensions were 380 mm 230 mm. The holograms' measurement grids included 32 11 measurement points along the x and y axes, respectively, at 4 cm of each other in both directions. These 352 measurement points covered, therefore, a hologram area of 128 cm 44 cm: a surface more than twice as large as the radiating structure. To improve the spatial resolution, the hologram was then expanded twice through zero-padding, before carrying out the double spatial Fourier transform, discussed in (10). An array, consisting of 8 pairs of Microflown pressure–velocity (PV) mini-probes, was used (Fig. 1c). Each pair included one probe capturing the intensity (pressure and particle velocity) in x direction and the other in y direction, on the hologram. Probes were rotated with respect to each other by a small angle ( 201) to avoid the acoustic shadowing in high frequencies. The pressure values, at each hologram grid point, used in the calculations are the results of the average of the pressures measured by the x and y oriented probes. The in-plane intensity components were obtained through cross-spectra GV x p and GV y p : ! ! !n !n I x ¼ 12 p v x ¼ 12 RefGV x p g and I y ¼ 12 p v y ¼ 12 RefGV y p g
(16)
These spectra were provided with a 32 channel OROS OR38 FFT analyser, with 64 spectral averaging.
Fig. 1. (a) Typical test setup used in all the three holography methods, namely, BAHIM, CIBNAH and NAH. (a) the structure and a probe array on the robot's arm. S1 and S2 show the active source apertures. Between S1 and S2 there are 3 supplementary apertures allowing us to vary the position of S2. (b) Transparent drawing of the same structure. (c) Pressure–velocity probes setup: double probe array on the robot's arm (BAHIM). (d) Pressure–velocity probes setup: single probe array on the robot's arm (CIBNAH, NAH).
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The sources (i.e., loudspeakers) emitted different signals: swept sine from the larger chamber and swept sine mixed with white noise from the smaller one. This combination enabled the generation of both coherent and partially coherent sound fields above the structure (Fig. 2).
6. BAHIM application and analysis Note: Since the hologram surface used in the calculations was more than 4 times the radiating surface, the mappings in this section, and in case of CIBNAH (Section 8), underwent different zoomings to better visualize the sources. Fig. 3 illustrates six mappings of the pressure on the surface of the structure, computed through back propagation of the complex pressure with the sound emitted from apertures 1 and 5 (14 cm away from each other). The complex pressure
Swept sine G +
White noise
G
Fig. 2. Signal generation setup. Varying the level of the white noise enables us to control the coherence between the two radiated signals.
Fig. 3. Pressure distribution mappings of the plate surface resulting from back propagation. Top to bottom: h ¼ 25 mm, 55 mm and 80 mm, respectively: (a) 800 Hz and (b) 1100 Hz. Dash–dot rectangles represent the actual surface of box top plate.
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distributions are associated with holograms at, from top to bottom, 25 mm, 55 mm and 80 mm above the plate surface. The sub-figure columns correspond to two arbitrary frequencies of 800 Hz (left) and 1100 Hz (right). These mappings demonstrate the ability of the method to detect the sources. A criterion that is not, however, sufficient. It is also important to verify whether we can faithfully highlight their spacing. The parallel dash–dot lines in Fig. 3 are to help this verification. It can be observed that as the distance h (hologram–structure) increases, the shape of the sources becomes less focused. In other words, less locally concentrated and their relative position more difficult to pinpoint. The source spacing, as measured from the mapping associated with h¼25 mm is 16 cm: a little different from the real spacing of 14 cm. In those resulting from h¼55 mm and 80 mm, the sources have a more scattered aspect. It is therefore harder to localize the pressure focal points. The approximate source spacing in these mappings varies from 16.2 cm to 26 cm (this last one at h ¼80 mm and at a frequency of 1100 Hz). These observations are in line with the discussion by Williams on the effects of the standoff distance h on the holographic reconstruction [9]. In case of BAHIM one can, however, pinpoint other (and even more important) causes for the above phenomena by considering that the complex pressure on the hologram is calculated from the in-plane intensities, i.e., in X and Y directions. Firstly, such probe orientations make them more sensitive to the unwanted signals (reflections and diffractions from surrounding objects and walls and even the investigated structures' parts). These intruding signals can clearly be seen at h¼ 55 mm and particularly at h ¼80 mm where in certain cases (e.g., Fig. 3b) they are so strong that they may interfere with the main source signal and obscure its position. Secondly, these components of the intensity decay more slowly than the normal intensity. This, in turn, leads to the slow decay of the sources as seen in the above figure. Hence, the closer the noise sources are with respect to each other, the more difficult it will be to distinguish them. It has been found, in the present investigation, that this difficulty can be reduced in certain cases by seeking at the regularization stage, an optimal width for the wavenumber filter applied to complex pressure in the wavenumber domain. Let us look at the example illustrated by Fig. 4. This is a case (800 Hz, h¼25 mm) where the sources are chosen to be 8 cm apart only (apertures 3 and 5 emitting). The two sub-figures differ with respect to the width of the k-filter: in the upper, the cut-off wavenumber in Kx direction is at 10/32 of the wavenumber range while the lower sub-figure corresponds to a larger width, i.e., 28/32 (cut-off wavenumber in Ky remains the same for both sub-figures). It can be noticed that widening the filter results in a more faithful rendition of the sources and their separation distance. This is because a wider filter includes more information for the back propagation calculations. The widening must, however, be reasonable, since beyond a certain width, the filter could also let undesired noise pass into the results. There is unfortunately no established rule to determine the optimal width of a filter. This technique can, therefore, be compromising in the presence of unwanted spectral components, e.g., noise, that the k-filter is exactly designed to reduce (or eliminate). The risk is higher as h increases, i.e., as the hologram backs away from the structure surface. Fig. 5 shows two pressure mappings at 1100 Hz, both related to the hologram at h¼ 80 mm. In the upper case, a k-filter width of 10/32 was used and in the lower, a slightly larger width of 14/32. One can observe a higher sensitivity to the surrounding noise (e.g., wall reflections, spurious noise leakages, etc.) at h¼ 80 mm, compared to the case h¼ 25 mm.
Fig. 4. Source separation by adjusting the k-filter width (800 Hz): (a) cut-off wavenumber at 10/32 of Kx range and (b) at 28/32 of Kx range. h ¼ 25 mm.
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Fig. 5. Effects of widening the k-filter width on longer hologram–structure distances (1100 Hz).
From the present investigation, one may conclude, about the BAHIM method, that
It is capable of detecting sources. Source localization precision drops with the increase in h. The pressure calculated on the surface of the source structure decays very slowly: a drawback especially when the sources are spatially close and/or when their amplitudes differ considerably.
In some cases, the above problem can be dealt with by widening the wavenumber filter applied to the measured spectra but there may then appear the risk of unwanted noise interference.
7. CIBNAH and NAH experimental setup CIBNAH has been tested on the same structure as illustrated in Fig. 1(a and b), and with the signal generation setup shown in Fig. 2. Also, the same signal processing procedures and equipment as mentioned in Section 5 were implemented. In particular, the cross and auto spectra Gpvz and Gvz vz (14) were averaged 64 times. Two sets of tests were performed corresponding to two holograms positioned at h ¼25 mm and 55 mm above the top plate. For these measurements, an array of 15 PU (pressure–velocity) mini-probes (Fig. 1d) was employed. The holograms' measurement grids included 30 15 points along the X and Y axes, respectively, and the measurement points were 3 cm apart in each direction. These 450 measurement points covered, therefore, a hologram area of 90 cm 45 cm, which is more than twice as large as the radiating surface of the box. Here also the hologram was expanded twice, before the spatial double Fourier transform. For the double-reference NAH, the same reference signals were used in both pressure and particle velocity hologram cases. These were the particle velocities provided by two PU mini-probes each placed adjacent to one of the active apertures. The signals represented the references to 450 double input–single output (hologram measurement points) systems. 8. CIBNAH application and analysis In this section, in addition to the previous performance criteria, i.e., source separation and localization, the results of CIBNAH will also be compared to those obtained from the double reference NAH, as mentioned earlier. The data associated with both methods have been obtained from the same acquisition. Fig. 6 represents the back propagated particle velocity mappings at 1100 Hz, calculated from CIBNAH (upper sub-figure) and classical NAH (lower sub-figure) with h¼30 mm. The sound was radiated from apertures 1 and 5 (distance 14 cm). The field was almost coherent with γ 2s1s2 ¼ 0:89, where γ 2s1s2 represents the coherence between the two sources (Fig. 7). Small amplitude differences, between the two methods, in this example and the following ones depend on the width of the wavenumber filters used during the back propagation calculations. One can observe a close similarity between CIBNAH and NAH results. Moreover, the sources are well separated and properly localized: their relative distance shown by the particle velocity maps is 14.5 cm, close to the real distance of 14 cm.
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Fig. 6. A comparison between: CIBNAH (a) and NAH (b) at 1100 Hz with sources separated by 14 cm: particle velocity on the structure surface associated with particle velocity measurements on the hologram. h ¼30 mm. Dash–dot rectangles represent the actual surface of the box top plate.
Fig. 7. Coherence between the sources (apertures 1 and 5).
In Fig. 8, another example of comparison between CIBNAH and NAH is illustrated. Here, the sound is emitted from apertures 3 and 5 (distanced by 8 cm) at 500 Hz and with h¼30 mm. Again there is good agreement between the CIBNAH back propagation calculations (upper sub-figure) and those of the NAH (lower sub-figure). Fairly accurate source separation of 8.25 cm is obtained from the CIBNAH particle velocity mapping. This leads us to consider CIBNAH as accurate as NAH, considering that the field was partially coherent ðγ 2s1s2 ¼ 0:8Þ. Regarding the effects of the variations of h (hologram–structure distance), in contrast with BAHIM (Fig. 3), CIBNAH yields quite repeatable results, as may be observed in Fig. 9 (500 Hz; γ 2s1s2 ¼ 0:8), the velocity mapping resulting from h¼30 mm and h¼55 mm are fairly similar in form and fairly similar in amplitude and source location. In the last example, let us look at a higher frequency case, i.e., 1400 Hz ðh ¼ 30 mmÞ; where the structure behaves in a rather more complicated manner. That is, apart from the apertures, other sources are present, e.g., the sound transmission across the plate's thickness and/or leakage from the edges. Also, the coherence between the sources at this frequency is as low as 0.72. A good agreement between CIBNAH (upper sub-figures in Fig. 10) and NAH (lower ones) can still be observed, even in such a complex field. The “cross-holography” U--4P (pressure field on the surface of the structure, from the measured hologram velocity field) has been obtained using propagator (12). The results for both CIBNAH and NAH are illustrated in the right hand side subfigures of Fig. 10, showing again a good agreement between the two.
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Fig. 8. Comparison between: CIBNAH (a) and NAH (b) at 500 Hz with sources separated by 8 cm: particle velocity on the structure surface, associated with particle velocity measurements on the hologram.
Fig. 9. Variation of h on the velocity back propagation at 500 Hz from the hologram h ¼30 mm (a); and h ¼55 mm (b).
9. Conclusions Two intensity-based holography methods have been discussed in this paper, using a sound radiating structure with controlled parameters. Originally, BAHIM was conceived to be applied with classical pressure–pressure intensity probes. In this investigation, however, the intensities are obtained using pressure–velocity probes. CIBNAH, a newly developed method, requires the latter kind of device as the complex velocity on the hologram is provided by the pressure–velocity spectral phase. In the present investigation, the problem of singularity in BAHIM has been addressed and resolved through approximation. Nevertheless, it has been found out that although the back propagation associated with this method enables us to detect sources on the surface of a structure, the accuracy in source separation cannot be considered as its strong point, although varying certain parameters such as k-filter width may occasionally lead to some improvement. BAHIM method can therefore be classified, in terms of the standards accuracy, in “expertise” category. In practical terms, it has the advantage of making use of pressure–pressure probes, which are very common nowadays. The described shortcomings of BAHIM have led to the development of CIBNAH whose common point with the former method is complex entity (particle velocity) calculations using intensity in order to avoid the need for reference signals.
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Fig. 10. Comparison between (a) CIBNAH and (b) NAH at 1400 Hz (sources separated by 14 cm). Left: particle velocity on the structure surface, associated with particle velocity measurements on the hologram; right: idem for pressure calculated from velocity. h ¼ 30 mm. Dash–dot rectangles represent the actual box radiating top plate.
This method enables us to calculate complex velocity on the hologram, and whence the velocity or pressure on the structure surface, which is another advantage compared to BAHIM. This new method has so far demonstrated good efficiency to detect sources and a fine precision in their localization. Its only limitation is equipment-wise, i.e., the need to make use of pressure–velocity probes not as common as pressure probes (i.e. microphones). Acknowledgement The author would like to thank Julien Marchand of the Occupational Noise Reduction lab, IET department of the INRS, for his valuable participation in the experimental parts of this investigation. References [1] D. Hallman, J.S. Bolton, A comparison of multi-reference nearfield acoustical holography procedures, Proceedings of Noise-Conference 94 (1994) 929–934. [2] M.A. Tomlinson, Partial source discrimination in near field acoustic holography, Applied Acoustics 57 (1999) 243–261. [3] Y.J. Kim, J.S. Bolton, H.S. Kwon, Partial sound field decomposition in multireference near-field acoustical holography by using optimally located virtual references, Journal of the Acoustical Society of America 115 (4) (2004) 1641–1652. [4] H.S. Kwon, J.S. Bolton, Partial field decomposition in nearfield acoustical holography by the use of singular value decomposition and partial coherence procedures, Proceedings of Noise-Conference 98 (1998) 649–654. [5] J. Hald, STSF – a unique technique for scan-based nearfield acoustical holography without restriction on coherence, B&K Technical Review, Vol. 1, 1988. [6] K.-U. Nam, Y.-H. Kim, A partial field decomposition algorithm and its examples for near-field acoustic holography, Journal of the Acoustical Society of America 116 (1) (2004) 172–185. [7] Th. Loyau, J.-C. Pascal, Broadband acoustic holography reconstruction from acoustic intensity measurements. I: principle of the method, Journal of the Acoustical Society of America 84 (5) (1988) 1744–1750. [8] J. Adin Mann III, J.-C. Pascal, Locating noise sources on an industrial air compressor using Broadband Acoustical Holography from Intensity Measurements (BAHIM), Noise Control Engineering Journal 39 (1) (1992) 3–12. [9] E.G. Williams, Fourier Acoustics, Academic Press, San Diego, London, Boston, New York, Sydney, Tokyo, Toronto, 1999. [10] Hadamard, Jacques, Sur les problèmes aux dérivées partielles et leur signification physique. Princeton University Bulletin, 1902, pp. 49–52. [11] F. Jacobsen, Y. Liu, Near field acoustic holography with particle velocity transducers, Journal of the Acoustical Society of America 118 (5) (2005) 3139–3144.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Reference-less acoustic holography techniques based on sound intensity A. Nejade n Institut National de Recherche et de Sécurité (INRS), 54519 Vandoeuvre Cedex, France
a r t i c l e in f o
abstract
Article history: Received 29 October 2013 Received in revised form 13 March 2014 Accepted 24 March 2014 Handling Editor: I. Lopez Arteaga Available online 29 April 2014
This paper describes a detailed implementation and application of the intensity-based holography method called BAHIM, and it investigates its possibilities and its limitations. Although this method is attractive in the sense that it bypasses the reference signal requirement, i.e., one of the main difficulties of the classical near-field acoustic holography, its degree of accuracy does not appear to be adequate. This constraint has consequently led to the development of another method called CIBNAH, based on complex intensity, as an alternative. This novel approach provides additional accuracy in detecting and localizing sources and good compatibility, in form and amplitude, with NAH results. & 2014 Elsevier Ltd. All rights reserved.
1. Introduction The application of classical near-field acoustic holography (NAH) to a structure requires a reference signal, when the sound field is generated by coherent sources. This application becomes more complex, however, when the sources are not or are partially coherent in which case they must each be represented by an ad hoc reference signal. This is a serious difficulty since in many cases, e.g., in machines, the sources are not necessarily identified: the identification being itself the preliminary aim of the holography. Numerous investigations have been carried out on systems that include partially coherent sources. The aim has been to decompose the total sound field into as many subfields as the sources. This task has been undertaken through several techniques such as ‘partial coherence method’ [1] or ‘virtual source method’ [2]. Hallman and Bolton remark that partial coherence decomposition would allow the partial fields to be associated with the fields radiated by individual sub-sources only if they are sufficiently separated spatially. This separation is not, however, described quantitatively. Application of the partial coherence technique requires a preliminary knowledge of the sources. ‘Virtual coherence and principle component analysis’ on the other hand is not particularly suitable to localize the sources [2]. This latter point is confirmed by Kim and Bolton [3]. Kwon et al. [4] present a comparison between singular value (introduced by Hald [5]) and partial coherence decomposition techniques and show that the reference signals must be captured at locations close to the individual real sources. This last condition is not always easy to satisfy (e.g., when source locations are unknown or inaccessible). To overcome such difficulties, some authors suggest the virtual reference approach [6] that in most cases consists of an iterative procedure where each step would result in more optimally placed and less numerous sources than the previous one. It is difficult to determine the number of such iterations, however, when the number and locations of the sources are unknown. Moreover, ending up with a number of virtual sources that exceeds that of the real sources could result in an
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inclusion of undesired noise source(s), as input(s), in the model. Kim and Bolton [3] undertake a more refined approach than that by Nam and Kim [6]. Here a number of virtual references are defined and then reduced by eliminating one between each pair of totally coherent sources. The optimized virtual sources are then treated through singular value decomposition (SVD) or partial coherence techniques to obtain partial fields associated with each. This technique again confirms the necessity that the virtual source locations be very close to the real sources, an exigency that may put the advantage of virtual reference technique into question. The above rather long, yet non-exhaustive, literature survey aimed at showing the complexity of implementation of the acoustic holography to the real physical systems, e.g., industrial machines: a persisting complexity, despite the numerous developments in the field. It is therefore desirable to search for alternative solutions that bypass the requirement for resorting to reference signals. During 1980s, Loyau and Pascal developed a method called broad band acoustic holography reconstruction from acoustic intensity measurements (BAHIM) [7] that could overcome the above complication since it did not need any reference signal. This method was accompanied by only a simple analytical example and was, later on, followed by a case history [8] without any theoretical or practical detail on the application. Although this method is fast and simple for practical applications, to the author's knowledge, it has not been followed by any further investigation. This paper presents an extended investigation of the BAHIM method of implementation, application on a structure as well as its limitations. It will be followed by the description of a newly developed broad band holography method called complex intensity based near-field acoustic holography (CIBNAH) that has proved to be more accurate than BAHIM, despite its theoretical simplicity. The common point between CIBNAH and BAHIM is the search for the phase of a complex entity (pressure or, in case of CIBNAH, pressure and/or particle velocity) on the hologram. They are, however, different from the theoretical and experimental points of view. Therefore, CIBNAH cannot be considered as an improved version, but as an alternative to BAHIM. 2. Theoretical background of BAHIM 2.1. Basic derivations ! The basic relation linking the pressure p to the active intensity vector I is given by 2 ! jpj ! γ; (1) I ¼ 2ρc ! ∇w ^ ∂w^ ∂w ^ ^ ^ ^ where ! γ ¼ k ¼ 1k ∂w ∂x i þ ∂y j þ ∂z k is the field factor, i; j et k are the unit vectors in x, y and z directions, respectively, ρ is the air density, c is the speed of sound, k is the acoustic wavenumber and w is the phase. Considering the components of the active intensity on the hologram plane, i.e., along the X and Y axes, relation (1) can be expressed as jpðx; yÞj2 ∂cðx; yÞ ^ ∂cðx; yÞ^ iþ j (2) I x ðx; yÞ^i þI y ðx; yÞ^j ¼ ∂x ∂y 2ρck The above relation can be written as i ∂cðx; yÞ^ ∂cðx; yÞ^ 2ρck h iþ j¼ I ðx; yÞ^i þ I y ðx; yÞ^j 2 x ∂x ∂y jpðx; yÞj
(3)
The implementation of relation (3), presented in space domain, is difficult. It is more practical when expressed in wavenumber kx et ky domain (corresponding to x and y directions) through spatial Fourier transformation: jkx ψðkx ; ky Þ^i jky ψðkx ; ky Þ^j ¼ k½ℱ fλx ðkx ; ky Þg^i þℱfλy ðkx ; ky Þg^j where ℱ represents the Fourier transform and 8 < λx ðkx ; ky Þ ¼ ℱ f2ρcI x ðx; yÞ=jpðx; yÞj2 g : λy ðkx ; ky Þ ¼ ℱ f2ρcIy ðx; yÞ=pðx; yÞ2 g
(4)
(5)
Using (4) and (5), the phase transform ψðkx ; ky Þ can be obtained as 2
2
2
2
ψðkx ; ky Þ ¼ jk½kx =ðkx þky Þλx ðkx ; ky Þ jk½ky =ðkx þky Þλy ðkx ; ky Þ
(6)
The phase in the spatial domain is then provided by the inverse spatial Fourier transform: ψ h ðx; yÞ ¼ ℱ 1 fψðkx ; ky Þg
(7)
Combination of this phase and the measured pressure amplitude then yields the complex pressure on the hologram: pffiffiffiffiffiffiffiffi where j ¼ 1:
ph ðx; yÞ ¼ jph ðx; yÞjejch ðx;yÞ
(8)
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2.2. Back propagation calculations Having obtained the complex pressures measured at all the grid points on the hologram located at a distance h from the surface of the studied structure, the usual holography procedures are applied using the following classical holography relation that calculates an entity γ (pressure, particle velocity) on the surface of the structure (xs ; ys ; zs ), from an entity χ measured on the hologram of coordinates (x; y; zh ) [9]: Z Z 1 γðxs ; ys ; zs Þ ¼ χðx; y; zh Þg p ðxs x; ys y; zs zh Þdx dy (9) 1
where gp represents the direct (pressure-pressure, velocity-velocity) or cross (pressure-velocity, velocity-pressure) green propagator. In most cases, it is easier to treat the above convolution in wavenumber domain, using Fourier transformation of (9) Γðkx ; ky ; zs Þ ¼ X ðkx ; ky ; zh Þ Gðkx ; ky ; zs zh Þ
(10)
where the terms Γ, X and G are the spatial double Fourier transforms of γ, χ and g, respectively. The expression for the propagator G depends on the measured entity on the hologram and the entity looked for through back propagation. In the direct holography (velocity back propagation U-U, pressure back propagation P-P), it is expressed as GUU ¼ GPP ¼ e jkz h ; (11) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 where h¼ zh zs and kz ¼ k kx ky . In case of “cross-holography”, the propagator takes other forms. For U-P, i.e., pressure calculation from velocity hologram measurements, used further in the article, the associated propagator has the form GUP ¼
ρck jkz h e : kz
(12)
To sum up, the application of BAHIM is carried out in two stages: (I) hologram complex pressure computation based on Eqs. (1)–(8) and (II) back propagation according to the classical NAH formulations expressed by relations (9) and (10). At the first stage of the computation (complex pressure), where a transformation from space to wavenumber domain is carried out, the usual classical holography procedures such as space windowing, zero-padding and k-space filtering do not seem necessary since all these techniques will be employed in the second stage, anyway. 2.3. Calculation process 2
2
2
2
The terms kx =ðkx þ ky Þ and ky =ðkx þ ky Þ in (6) are dependent on the interaction between the x and y components of the wavenumbers, i.e., kx ¼ 2πðnx =Lx Þ et ky ¼ 2πðny =Ly Þ where nx et ny are the wavenumber indices, represented in vector shape as: nx ¼[0, 1, 2, 3, …, ((mx =2) 1), 0, ((mx =2) 1), ((mx =2) 2), …, 3, 2, 1] and ny ¼[0, 1, 2, 3, …, ((my =2) 1), 0, ((my =2) 1), (my =2) 2), …, 3, 2, 1]0 , where mx et my are the measurement point numbers along kx and ky axes, respectively, and the sign 0 implies the transpose of the vector. One can observe that for certain values of ðnx , ny Þ, i.e. (1, 1), (1, mx/2), (my/2, 1) and (mx/2, my/2), the wavenumbers kx and ky simultaneously vanish, leading to 4 singularity points in (6). An example of such a situation is presented in Table 1, showing an 8 8 wavenumber matrix where singularities are designated by “undefined”. As mentioned earlier, these singularities occur at kx ¼ky ¼0, pointing to a plane wave that propagates uniquely in the z direction (i.e., perpendicular to the hologram). It means that the in-plane phase gradients are equal to zero. Physically speaking, this implies a DC offset, in the phase, whose value is undefined. We are, therefore, in the presence of an ill-posed problem since it does not satisfy the uniqueness of solution [10]. This represents the main computational difficulty in the application of BAHIM. As in all ill-conditioned problems, one must search for approximate solutions. We may replace the singularities either by the average value of the adjacent cells or by zeros following the behaviour of the associated columns. In both cases the singularity peaks disappear allowing all the other measurement points on the hologram to influence the
Table 1 An example of the matrix jkK x =ðK 2x þ K 2y Þ. Undefined 0.000E þ00 0.000E þ00 0.000E þ00 Undefined 0.000E þ00 0.000E þ00 0.000E þ00
0 0.7463i 0 0.3731i 0 0.1492i 0 0.0746i 0 0.7463i 0 0.0746i 0 0.1492i 0 0.3731i
0 0.3731i 0 0.2985i 0 0.1865i 0 0.1148i 0 0.3731i 0 0.1148i 0 0.1865i 0 0.2985i
0 0.2487i 0 0.2239i 0 0.1722i 0 0.1243i 0 0.2487i 0 0.1243i 0 0.1722i 0 0.2239i
Undefined 0.000E þ00 0.000E þ00 0.000E þ00 Undefined 0.000E þ00 0.000E þ00 0.000E þ00
0 þ0.2487i 0 þ0.2239i 0 þ0.1722i 0 þ0.1243i 0 þ0.2487i 0 þ0.1243i 0 þ0.1722i 0 þ0.2239i
0 þ0.3731i 0 þ0.2985i 0 þ0.1865i 0 þ0.1148i 0 þ0.3731i 0 þ0.1148i 0 þ0.1865i 0 þ0.2985i
0 þ0.7463i 0 þ0.37317i 0 þ0.1492i 0 þ0.0746i 0 þ0.7463i 0 þ0.0746i 0 þ0.1492 0 þ0.3731i
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final result in proportion to their amplitude. The consequence of the back propagation computation will be the same for both types of approximations provided that Npm⪢4, where Npm is the total number of the measurement points and 4 refers 2 2 to the number of singularity points in the jkkx =ðkx þ ky Þ matrix. 3. Complex intensity based near-field acoustic holography (CIBNAH), a novel method of reference-less holography, as alternative to BAHIM 3.1. Introduction BAHIM method was developed as an alternative allowing the bypass of the main difficulty of classical (or multireference) holography, namely, the choice of reference signal(s). To do so, the theoretical derivations aim at calculating the pressure phase distribution on the hologram, the information that, otherwise, can only be provided by a reference signal. However, as it will be demonstrated in Section 5, this method shows inconsistencies in its accuracy mainly because of the slow decay of the in-plane (x, y) components of the intensity, from which the phase distribution on the hologram is deduced. 3.2. CIBNAH theoretical background Holography with no need for reference signal(s) can otherwise be achieved in a simpler manner, by using the normal complex intensity obtained by the following basic and direct relation: ! !n I z ¼ 12 p v z
(13) ! n where v z represents normal particle velocity vector and its complex conjugate. CIBNAH consists of using the phase of this complex intensity along with the measured particle velocity (or pressure) module in spectral domain. This phase has essentially the same nature as the one calculated by BAHIM. Pressure–velocity probes easily available nowadays in the market allow simultaneous measurements of the two quantities (at the same point in space, up to 12 kHz) and consequently their relative phase. As mentioned in some works in the literature [11], a particle velocity field generated by a source undergoes a much faster decay in its plane than the pressure field. This leads to a better source detection and separation in the back propagation. The same observation was made during our investigations. For this reason, the focus, in the present work, has been on velocity–velocity or velocity–pressure holography, i.e., particle velocity back propagation using particle velocity hologram measurements. From (13), the complex particle velocity can be expressed in frequency domain as v~ z ¼
pffiffiffiffiffiffiffiffiffiffi Gpvz pffiffiffiffiffiffiffiffiffiffi ¼ Gvz vz ejch Gvz vz Gpvz
(14)
where G's represent the auto and cross-spectra corresponding to the entities p (pressure) and vz (normal particle velocity), both acquired simultaneously on the measurement plane (hologram) using a pressure–velocity probe. Sign represents a pffiffiffiffiffiffiffiffiffiffi complex value. The two terms Gpvz and Gvz vz , in the above formula, are therefore the outcomes of simultaneous spectral averages. The phase ch between the pressure and the particle velocity includes information both on spatial coordinates of the corresponding measurement point (on the hologram) and the eventual source signal fluctuations of the investigated structure. An alternative representation for (14) can be given by qffiffiffiffiffiffiffiffi Gpvz qffiffiffiffiffiffiffiffi v~ z ¼ Gpp ¼ H pvz Gpp ; (15) Gpp where the transfer function H pvz may provide a more consistent (and stable) phase for the complex velocity. As mentioned earlier, the lack of a need for reference signal(s) constitutes the basic advantage of CIBNAH over NAH, and especially the multi-reference NAH. There are also other potential gains in applying CIBNAH, namely,
Considerably lower number of the terms involved in the CIBNAH formulation leading to less possible computation errors. Inaccuracies due to approximate (or inappropriate) choice of reference signals required by NAH are absent. The experimental and computational processes are easier and faster. 4. Multi-reference NAH In several cases below, the results of CIBNAH will be compared to those of double reference near-field acoustic holography (NAH), using partial coherence techniques. This latter approach was undertaken since the sources were partially coherent. The data for both techniques (CIBNAH and NAH) were acquired simultaneously. For this reason, and because we are investigating strictly the same structure, the results from the two methods must be analogous. This comparison aims at verifying the accuracy of CIBNAH and also to show its insensitivity to the value of the coherence between the sources.
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5. BAHIM test setup A special structure was designed to test the method in a controlled manner. It is a 38 23 28 cm3 double chamber wooden box, each chamber half-full of absorbing materials and containing a source (loudspeaker) at its mid height. Circular apertures have been drilled on the top plate: one communicating with the smaller chamber and four with the larger one (Fig. 1a, b). This configuration enables us to vary certain testing parameters such as the distance between the sources while knowing their exact position (by obstructing three out of four apertures on the larger chamber). Results presented in this paper correspond to cases where the sound was emitted through the apertures 1 and 5 (14 cm source spacing) and through 3 and 5 (8 cm spacing). Also, through the choice of the input signals, coherence between the sources could be varied. Three sets of tests were performed with measurement surface (hologram) parallel to and above the box top plate at distances (h) of 25 mm, 55 mm (and 80 mm for BAHIM, uniquely). The top plate's dimensions were 380 mm 230 mm. The holograms' measurement grids included 32 11 measurement points along the x and y axes, respectively, at 4 cm of each other in both directions. These 352 measurement points covered, therefore, a hologram area of 128 cm 44 cm: a surface more than twice as large as the radiating structure. To improve the spatial resolution, the hologram was then expanded twice through zero-padding, before carrying out the double spatial Fourier transform, discussed in (10). An array, consisting of 8 pairs of Microflown pressure–velocity (PV) mini-probes, was used (Fig. 1c). Each pair included one probe capturing the intensity (pressure and particle velocity) in x direction and the other in y direction, on the hologram. Probes were rotated with respect to each other by a small angle ( 201) to avoid the acoustic shadowing in high frequencies. The pressure values, at each hologram grid point, used in the calculations are the results of the average of the pressures measured by the x and y oriented probes. The in-plane intensity components were obtained through cross-spectra GV x p and GV y p : ! ! !n !n I x ¼ 12 p v x ¼ 12 RefGV x p g and I y ¼ 12 p v y ¼ 12 RefGV y p g
(16)
These spectra were provided with a 32 channel OROS OR38 FFT analyser, with 64 spectral averaging.
Fig. 1. (a) Typical test setup used in all the three holography methods, namely, BAHIM, CIBNAH and NAH. (a) the structure and a probe array on the robot's arm. S1 and S2 show the active source apertures. Between S1 and S2 there are 3 supplementary apertures allowing us to vary the position of S2. (b) Transparent drawing of the same structure. (c) Pressure–velocity probes setup: double probe array on the robot's arm (BAHIM). (d) Pressure–velocity probes setup: single probe array on the robot's arm (CIBNAH, NAH).
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The sources (i.e., loudspeakers) emitted different signals: swept sine from the larger chamber and swept sine mixed with white noise from the smaller one. This combination enabled the generation of both coherent and partially coherent sound fields above the structure (Fig. 2).
6. BAHIM application and analysis Note: Since the hologram surface used in the calculations was more than 4 times the radiating surface, the mappings in this section, and in case of CIBNAH (Section 8), underwent different zoomings to better visualize the sources. Fig. 3 illustrates six mappings of the pressure on the surface of the structure, computed through back propagation of the complex pressure with the sound emitted from apertures 1 and 5 (14 cm away from each other). The complex pressure
Swept sine G +
White noise
G
Fig. 2. Signal generation setup. Varying the level of the white noise enables us to control the coherence between the two radiated signals.
Fig. 3. Pressure distribution mappings of the plate surface resulting from back propagation. Top to bottom: h ¼ 25 mm, 55 mm and 80 mm, respectively: (a) 800 Hz and (b) 1100 Hz. Dash–dot rectangles represent the actual surface of box top plate.
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distributions are associated with holograms at, from top to bottom, 25 mm, 55 mm and 80 mm above the plate surface. The sub-figure columns correspond to two arbitrary frequencies of 800 Hz (left) and 1100 Hz (right). These mappings demonstrate the ability of the method to detect the sources. A criterion that is not, however, sufficient. It is also important to verify whether we can faithfully highlight their spacing. The parallel dash–dot lines in Fig. 3 are to help this verification. It can be observed that as the distance h (hologram–structure) increases, the shape of the sources becomes less focused. In other words, less locally concentrated and their relative position more difficult to pinpoint. The source spacing, as measured from the mapping associated with h¼25 mm is 16 cm: a little different from the real spacing of 14 cm. In those resulting from h¼55 mm and 80 mm, the sources have a more scattered aspect. It is therefore harder to localize the pressure focal points. The approximate source spacing in these mappings varies from 16.2 cm to 26 cm (this last one at h ¼80 mm and at a frequency of 1100 Hz). These observations are in line with the discussion by Williams on the effects of the standoff distance h on the holographic reconstruction [9]. In case of BAHIM one can, however, pinpoint other (and even more important) causes for the above phenomena by considering that the complex pressure on the hologram is calculated from the in-plane intensities, i.e., in X and Y directions. Firstly, such probe orientations make them more sensitive to the unwanted signals (reflections and diffractions from surrounding objects and walls and even the investigated structures' parts). These intruding signals can clearly be seen at h¼ 55 mm and particularly at h ¼80 mm where in certain cases (e.g., Fig. 3b) they are so strong that they may interfere with the main source signal and obscure its position. Secondly, these components of the intensity decay more slowly than the normal intensity. This, in turn, leads to the slow decay of the sources as seen in the above figure. Hence, the closer the noise sources are with respect to each other, the more difficult it will be to distinguish them. It has been found, in the present investigation, that this difficulty can be reduced in certain cases by seeking at the regularization stage, an optimal width for the wavenumber filter applied to complex pressure in the wavenumber domain. Let us look at the example illustrated by Fig. 4. This is a case (800 Hz, h¼25 mm) where the sources are chosen to be 8 cm apart only (apertures 3 and 5 emitting). The two sub-figures differ with respect to the width of the k-filter: in the upper, the cut-off wavenumber in Kx direction is at 10/32 of the wavenumber range while the lower sub-figure corresponds to a larger width, i.e., 28/32 (cut-off wavenumber in Ky remains the same for both sub-figures). It can be noticed that widening the filter results in a more faithful rendition of the sources and their separation distance. This is because a wider filter includes more information for the back propagation calculations. The widening must, however, be reasonable, since beyond a certain width, the filter could also let undesired noise pass into the results. There is unfortunately no established rule to determine the optimal width of a filter. This technique can, therefore, be compromising in the presence of unwanted spectral components, e.g., noise, that the k-filter is exactly designed to reduce (or eliminate). The risk is higher as h increases, i.e., as the hologram backs away from the structure surface. Fig. 5 shows two pressure mappings at 1100 Hz, both related to the hologram at h¼ 80 mm. In the upper case, a k-filter width of 10/32 was used and in the lower, a slightly larger width of 14/32. One can observe a higher sensitivity to the surrounding noise (e.g., wall reflections, spurious noise leakages, etc.) at h¼ 80 mm, compared to the case h¼ 25 mm.
Fig. 4. Source separation by adjusting the k-filter width (800 Hz): (a) cut-off wavenumber at 10/32 of Kx range and (b) at 28/32 of Kx range. h ¼ 25 mm.
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Fig. 5. Effects of widening the k-filter width on longer hologram–structure distances (1100 Hz).
From the present investigation, one may conclude, about the BAHIM method, that
It is capable of detecting sources. Source localization precision drops with the increase in h. The pressure calculated on the surface of the source structure decays very slowly: a drawback especially when the sources are spatially close and/or when their amplitudes differ considerably.
In some cases, the above problem can be dealt with by widening the wavenumber filter applied to the measured spectra but there may then appear the risk of unwanted noise interference.
7. CIBNAH and NAH experimental setup CIBNAH has been tested on the same structure as illustrated in Fig. 1(a and b), and with the signal generation setup shown in Fig. 2. Also, the same signal processing procedures and equipment as mentioned in Section 5 were implemented. In particular, the cross and auto spectra Gpvz and Gvz vz (14) were averaged 64 times. Two sets of tests were performed corresponding to two holograms positioned at h ¼25 mm and 55 mm above the top plate. For these measurements, an array of 15 PU (pressure–velocity) mini-probes (Fig. 1d) was employed. The holograms' measurement grids included 30 15 points along the X and Y axes, respectively, and the measurement points were 3 cm apart in each direction. These 450 measurement points covered, therefore, a hologram area of 90 cm 45 cm, which is more than twice as large as the radiating surface of the box. Here also the hologram was expanded twice, before the spatial double Fourier transform. For the double-reference NAH, the same reference signals were used in both pressure and particle velocity hologram cases. These were the particle velocities provided by two PU mini-probes each placed adjacent to one of the active apertures. The signals represented the references to 450 double input–single output (hologram measurement points) systems. 8. CIBNAH application and analysis In this section, in addition to the previous performance criteria, i.e., source separation and localization, the results of CIBNAH will also be compared to those obtained from the double reference NAH, as mentioned earlier. The data associated with both methods have been obtained from the same acquisition. Fig. 6 represents the back propagated particle velocity mappings at 1100 Hz, calculated from CIBNAH (upper sub-figure) and classical NAH (lower sub-figure) with h¼30 mm. The sound was radiated from apertures 1 and 5 (distance 14 cm). The field was almost coherent with γ 2s1s2 ¼ 0:89, where γ 2s1s2 represents the coherence between the two sources (Fig. 7). Small amplitude differences, between the two methods, in this example and the following ones depend on the width of the wavenumber filters used during the back propagation calculations. One can observe a close similarity between CIBNAH and NAH results. Moreover, the sources are well separated and properly localized: their relative distance shown by the particle velocity maps is 14.5 cm, close to the real distance of 14 cm.
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Fig. 6. A comparison between: CIBNAH (a) and NAH (b) at 1100 Hz with sources separated by 14 cm: particle velocity on the structure surface associated with particle velocity measurements on the hologram. h ¼30 mm. Dash–dot rectangles represent the actual surface of the box top plate.
Fig. 7. Coherence between the sources (apertures 1 and 5).
In Fig. 8, another example of comparison between CIBNAH and NAH is illustrated. Here, the sound is emitted from apertures 3 and 5 (distanced by 8 cm) at 500 Hz and with h¼30 mm. Again there is good agreement between the CIBNAH back propagation calculations (upper sub-figure) and those of the NAH (lower sub-figure). Fairly accurate source separation of 8.25 cm is obtained from the CIBNAH particle velocity mapping. This leads us to consider CIBNAH as accurate as NAH, considering that the field was partially coherent ðγ 2s1s2 ¼ 0:8Þ. Regarding the effects of the variations of h (hologram–structure distance), in contrast with BAHIM (Fig. 3), CIBNAH yields quite repeatable results, as may be observed in Fig. 9 (500 Hz; γ 2s1s2 ¼ 0:8), the velocity mapping resulting from h¼30 mm and h¼55 mm are fairly similar in form and fairly similar in amplitude and source location. In the last example, let us look at a higher frequency case, i.e., 1400 Hz ðh ¼ 30 mmÞ; where the structure behaves in a rather more complicated manner. That is, apart from the apertures, other sources are present, e.g., the sound transmission across the plate's thickness and/or leakage from the edges. Also, the coherence between the sources at this frequency is as low as 0.72. A good agreement between CIBNAH (upper sub-figures in Fig. 10) and NAH (lower ones) can still be observed, even in such a complex field. The “cross-holography” U--4P (pressure field on the surface of the structure, from the measured hologram velocity field) has been obtained using propagator (12). The results for both CIBNAH and NAH are illustrated in the right hand side subfigures of Fig. 10, showing again a good agreement between the two.
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Fig. 8. Comparison between: CIBNAH (a) and NAH (b) at 500 Hz with sources separated by 8 cm: particle velocity on the structure surface, associated with particle velocity measurements on the hologram.
Fig. 9. Variation of h on the velocity back propagation at 500 Hz from the hologram h ¼30 mm (a); and h ¼55 mm (b).
9. Conclusions Two intensity-based holography methods have been discussed in this paper, using a sound radiating structure with controlled parameters. Originally, BAHIM was conceived to be applied with classical pressure–pressure intensity probes. In this investigation, however, the intensities are obtained using pressure–velocity probes. CIBNAH, a newly developed method, requires the latter kind of device as the complex velocity on the hologram is provided by the pressure–velocity spectral phase. In the present investigation, the problem of singularity in BAHIM has been addressed and resolved through approximation. Nevertheless, it has been found out that although the back propagation associated with this method enables us to detect sources on the surface of a structure, the accuracy in source separation cannot be considered as its strong point, although varying certain parameters such as k-filter width may occasionally lead to some improvement. BAHIM method can therefore be classified, in terms of the standards accuracy, in “expertise” category. In practical terms, it has the advantage of making use of pressure–pressure probes, which are very common nowadays. The described shortcomings of BAHIM have led to the development of CIBNAH whose common point with the former method is complex entity (particle velocity) calculations using intensity in order to avoid the need for reference signals.
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Fig. 10. Comparison between (a) CIBNAH and (b) NAH at 1400 Hz (sources separated by 14 cm). Left: particle velocity on the structure surface, associated with particle velocity measurements on the hologram; right: idem for pressure calculated from velocity. h ¼ 30 mm. Dash–dot rectangles represent the actual box radiating top plate.
This method enables us to calculate complex velocity on the hologram, and whence the velocity or pressure on the structure surface, which is another advantage compared to BAHIM. This new method has so far demonstrated good efficiency to detect sources and a fine precision in their localization. Its only limitation is equipment-wise, i.e., the need to make use of pressure–velocity probes not as common as pressure probes (i.e. microphones). Acknowledgement The author would like to thank Julien Marchand of the Occupational Noise Reduction lab, IET department of the INRS, for his valuable participation in the experimental parts of this investigation. References [1] D. Hallman, J.S. Bolton, A comparison of multi-reference nearfield acoustical holography procedures, Proceedings of Noise-Conference 94 (1994) 929–934. [2] M.A. Tomlinson, Partial source discrimination in near field acoustic holography, Applied Acoustics 57 (1999) 243–261. [3] Y.J. Kim, J.S. Bolton, H.S. Kwon, Partial sound field decomposition in multireference near-field acoustical holography by using optimally located virtual references, Journal of the Acoustical Society of America 115 (4) (2004) 1641–1652. [4] H.S. Kwon, J.S. Bolton, Partial field decomposition in nearfield acoustical holography by the use of singular value decomposition and partial coherence procedures, Proceedings of Noise-Conference 98 (1998) 649–654. [5] J. Hald, STSF – a unique technique for scan-based nearfield acoustical holography without restriction on coherence, B&K Technical Review, Vol. 1, 1988. [6] K.-U. Nam, Y.-H. Kim, A partial field decomposition algorithm and its examples for near-field acoustic holography, Journal of the Acoustical Society of America 116 (1) (2004) 172–185. [7] Th. Loyau, J.-C. Pascal, Broadband acoustic holography reconstruction from acoustic intensity measurements. I: principle of the method, Journal of the Acoustical Society of America 84 (5) (1988) 1744–1750. [8] J. Adin Mann III, J.-C. Pascal, Locating noise sources on an industrial air compressor using Broadband Acoustical Holography from Intensity Measurements (BAHIM), Noise Control Engineering Journal 39 (1) (1992) 3–12. [9] E.G. Williams, Fourier Acoustics, Academic Press, San Diego, London, Boston, New York, Sydney, Tokyo, Toronto, 1999. [10] Hadamard, Jacques, Sur les problèmes aux dérivées partielles et leur signification physique. Princeton University Bulletin, 1902, pp. 49–52. [11] F. Jacobsen, Y. Liu, Near field acoustic holography with particle velocity transducers, Journal of the Acoustical Society of America 118 (5) (2005) 3139–3144.