RT-NAH : A NEARFIELD ACOUSTIC HOLOGRAPHY TECHNIQUE FOR REAL-TIME RECONSTRUCTION Jean-Hugh Thomas, Vincent Grulier, Sébastien Paillasseur and Jean-Claude Pascal Laboratoire d'Acoustique de l'Université du Maine (CNRS UMR 6613) and Ecole Nationale Supérieure d'Ingénieurs du Mans (ENSIM) Université du Maine, rue Aristote, 72000 Le Mans, France. [email protected], [email protected]

ABSTRACT Instead of the generally named "time holography" techniques which operate in the frequency domain to process differently progressive and evanescent waves, the Real-Time Nearfield Acoustic Holography algorithm (RT-NAH) works in a time-wavenumber domain. Each point of the instantaneous wavenumber spectrum is convolved by an numerical filter. Thus, it is possible to continuously reconstruct the acoustic pressure field on the plane from where some non stationary sources radiate. The starting point is the acquisition, in the nearfield of the sources, of a pressure field which fluctuates in time. Then a space Fourier transform is applied to each temporal sample of the whole antenna of microphones. An analytic study of the problem shows that the pressure field can be propagated on a forward plane by filtering it in the time-wavenumber domain. In order to backpropagating the pressure field to the source plane, the impulse response has to be inverted. The realization of the inverse numerical filter is reported. After filtering, the fluctuating pressure signal is obtained by computing the inverse space Fourier transform of each time sample of the instantaneous wavenumber spectrum rebuilt on the source plane.

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INTRODUCTION

The beamforming technique makes it possible to localize fluctuating acoustic sources by simple processing, providing an instantaneous signal associated to the source located at the focal point. However this technique has the disadvantage of having a resolution which decreases dramatically in the low frequencies. Because of the use of the evanescent waves, Nearfield Acoustic Holography (NAH) preserves an excellent resolution in the low frequencies [1]. It is then necessary to work in the frequency-wavenumber domain in which a separate processing of progressive and evanescent waves is feasible. The standard NAH algorithm limits this method to stationary sources. The “time methods” proposed for acoustic holography always use the same algorithm but for each component of the FFT of the time signal before making the inverse Fourier transform [2], [3], [4]. To use acoustic holography on the fluctuating sources like the moving mechanisms (windscreen wiper) or on sources whose spectral components evolve (run-up test of rotating machinery), a new algorithm named Real-Time Nearfield Acoustic Holography (RT-NAH) is proposed [5]. Contrary to the previous time methods, RT-NAH works directly in the time-wavenumber domain and requires no Fourier transform of the time signals. First, the principle of standard NAH in frequency domain and its time extension are discussed. The paper focuses on specificities of this new RT-NAH method involving inverse filtering in the time-wavenumber domain to rebuild continuously the signal at any point of the source, taking into account the nearfield components. 2 2.1

PLANAR NEARFIELD ACOUSTIC HOLOGRAPHY Frequency methods for stationary field

The aim of planar NAH is to compute the pressure field on the source plane z = z s from pressure measurements done from a plane z = z a . When the sources of sound are stationary, the pressure field is acquired from z = z a at different moments, from the displacement of a small microphone array using a two-axis robot. Then, the phase relationships between the pressure measurements are recovered from cross-spectrum processing using several reference signals and a Principal Component Analysis. It is usual to then extract the spatial pressure field p( x, y, z a , ω 0 ) from the measurement surface for a chosen angular frequency ω 0 . Solving the inverse problem provides the spatial pressure field p( x, y, z s , ω 0 ) for ω 0 on the source plane (see Fig. 1). It is done by applying in the wavenumber domain an inverse propagator G −1 which depends on the nature of the waves (Eq. 1). Indeed, the wavenumber domain separates the plane waves from the evanescent waves which decay exponentially: the wavenumber components k x and k y lie within the radiation circle of radius k = ω 0 c for plane waves ( c denotes the sound celerity) whereas they fall outside the circle for evanescent waves P (k x , k y , z s , ω ) = P(k x , k y , z a , ω ) G −1 (k x , k y , z a − z s , ω ) where, for plane progressive waves and for evanescent waves, respectively

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(1)

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e j k 2 − k x2 − k y2 ( za − z S ) if k x2 + k y2 ≤ k 2 (progressive)  (2) G (k x , k y , z a − z s , ω ) =  2 2 2 k +k −k ( za − zS ) if k x2 + k y2 > k 2 (evanescente)  e x y A critical point in the reconstruction concerns the amplification of evanescent waves. To limit the distortion due to the presence of noise, some techniques of filtering and regularization [6] in the wavenumber domain are used. A two dimensional inverse Fourier transform of Eq. 1 is used to obtain the spatial pressure field p( x, y, z s , ω 0 ) on the source plane. Therefore, the great advantage of NAH is to provide an image of the spatial pressure field for one frequency or a frequency band that allows us to locate the parts of the system studied which radiate the most. However, monitoring a system in order to detect any defects or predict the presence of inappropriate conditions requires inspection of the time evolution of some features designed to highlight the behaviour of the system. When the statistical properties of the emitted acoustic signals fluctuate in time, standard NAH is unsuitable because the spatial pressure field obtained by the method fluctuates in time whatever the frequency studied. −1

2.2

Time methods using frequency-wavenumber processing

A time-method proposed by Deblauwe et al. operates a Fourier transform on a transient time signal applying NAH processing to each frequency raw before returning in time domain using an inverse Fourier transform [2]. The use of temporal Fourier transforms makes the method effective only for short time events. Another approach, proposed by Hald [3], [4], provides for a chosen frequency band, a time dependent mapping of the spatial pressure field. This representation is due to the use of a Short Time Fourier Transform on the temporal acoustic signals acquired by the microphone array.

Fig. 1. Synopsis of NAH processing for stationary sources (a) and RT-NAH processing for non stationary sources (b) (2D FTx,y : spatial two dimensional Fourier transform).

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REAL-TIME NEARFIELD ACOUSTIC HOLOGRAPHY (RT-NAH)

3.1

Theory

The starting point is the wave-equation in plane geometry (Eq. 3) on which a two-dimension Fourier transform, with respect to space, is operated, yielding Eq. 4 1 ∂ 2 p ( x, y , z , t ) ∇ 2 p ( x, y , z , t ) − 2 = 0, (3) c ∂t 2 ∂ 2 P (k x , k y , z , t )

2 1 ∂ P (k x , k y , z , t ) − 2 − k x2 − k y2 P(k x , k y , z , t ) = 0 . (4) 2 2 ∂z c ∂t The pressure field P (k x , k y , z s , t ) on the plane source in a time-wavenumber domain is the

solution of Eq. 4 [7]

(

)

P (k x , k y , z s , t ) = P (k x , k y , z a , t ) ∗ h −1 (k x , k y , ∆z , t )

(5)

where h −1 (k x , k y , ∆z , t ) is an inverse impulse response operating in the time-wavenumber domain to back-propagate the pressure field from the measurement plane to the source plane ( ∆z = z a − z s ). Since the objective here is to provide a time-dependent pressure signal on the source plane, the Real-Time Nearfield Acoustic Holography may be described by the following equation (see the synopsis of the method in Fig. 1b)

{

}

p( x, y, z s , t ) = Fx−, 1y Fx , y {p( x, y, z a , t )}∗ h −1 (k x , k y , ∆z , t )

(6)

Before defining h −1 (k x , k y , ∆z , t ) , let us give the expression of the impulse response h(k x , k y , ∆z , t ) .

3.2

The impulse response for forward propagating

The impulse response h(k x , k y , ∆z , t ) , whose goal is to project the acoustic pressure field

from the source plane to the measurement plane, depends, of course, on the propagation distance ∆z = z a − z s . It operates in the time-wavenumber domain. The analytical expression below is also given in [8], [9] ∆z 2  ∆z 2  ∆z   ∆z   2 2 2 2 2 2 t − 2 h(k x , k y , ∆z , t ) = δ  t −  − ∆z k x + k y Γ t −  J 1 c k x + k y t − 2 c  c   c  c   (7) J1 denotes the first order Bessel function, δ (t ) the Dirac delta function and Γ(t ) the Heaviside step function. Thus P (k x , k y , z a , t ) , the time-dependent wavenumber spectrum on the measurement plane is deduced from that on the source plane P (k x , k y , z s , t ) using the following equation P (k x , k y , z a , t ) = P(k x , k y , z s , t ) ∗ h(k x , k y , ∆z , t ) (8)

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INVERSE FILTERING FOR THE REAL-TIME METHOD

4.1

The direct filter for the forward propagation

The aim of the approach is, first, to propose a causal digital filter with a finite-duration impulse response (FIR) hd (k x , k y , ∆z , n ) to solve the direct problem, then to design the inverse filter with an impulse response hd−1 (k x , k y , ∆z , n ) . The feasibility study of this approach for the direct problem is reported in [7] where the analytic impulse response of Eq. 7 is converted into a finite sequence of numbers. Grulier in [8] investigates several methods to sample the impulse response : the method which numerically computes the convolution of the impulse response by a Kaiser-Bessel window using the trapezoidal formula gives the best result. 4.2

The inverse filter for backward propagating

The final stage must provide the inverse impulse response hd−1 (k x , k y , ∆z , n ) to fulfill the

following condition

hd−1 (k x , h y , ∆z , n ) ∗ hd (k x , h y , ∆z , n ) = δ (n ) .

(9)

The use of an equalizer by adaptive filtering [10] is selected to solve this problem. Thus, given the z-transform H d ( z ) of the discrete sequence hd [n] , the aim is to find the z-transform H d−1 (z ) so that Y ( z ) = H d−1 ( z ) H d (z )U ( z ) = U ( z ) where U ( z ) and Y ( z ) respectively denote the z-transforms of the input and the output of the system. The searched solution hd−1 [n] is obtained by minimizing the mean-squared criterion

(

)

(

)

J = ∑n= −∞ δ [n] − hd [n ] ∗ hd−1 [n ] =∑n = −∞ δ [n ] − ∑m =0 hd−1 [m] hd [n − m] . +∞

+∞

2

N −1

2

(10)

The solution sought corresponds to the impulse response of a FIR filter with hd−1 [n] = 0 for n < 0 and n > N . The minimization of criterion J gives the solution in a matrix formalism h −d1 = C −hd1 h rd , where h

−1 d

is the vector of the solutions, C

−1 hd

(11)

the inverse correlation matrix of the direct filter

and h the reverse vector of the direct filter ( hd [n] = hdr [− n] ). r d

Real-Time Nearfield Holography succeeds in providing time continuous signals on each node of a virtual area on the source plane that is the image of the measurement surface. It is as if the time measurements had been made directly on the surface, including the acoustic sources. A simulation example involving three point sources radiating frequency modulated signals at z = 0 and an antenna of 17 × 17 microphones ( 1 m × 1 m ) at z a = 0.157 m is highlighted in Fig. 2 with three signals: the one obtained from the measurement plane, the reconstructed time-dependent pressure signal on the plane z = 0.05 m and the reference signal observed at the same position. The signal on the left has distortion due to the influence of the

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other monopoles. The distortion disappears on the back-propagated signal on the middle. The pattern is similar to the signal emitted by the monopole on the right of the Fig. 2.

Fig. 2. Simulation of inverse filtering to reconstruct the acoustic signal at a point of the source plane (the acquired signal on the left, the reconstructed on the middle and the reference on the right).

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CONCLUSIONS

A technique called Real-Time Acoustic Holography based on the use of an inverse digital filter in the wavenumber domain is proposed. From the acquisition of the acoustic field in the near-field of the sources, the method provides time continuous reconstructed signals directly on the source plane. The approach seems promising even though additional work is needed. Indeed the continuous emission rebuilt on a point of a grid area on the source plane may then be used for diagnosis purposes. REFERENCES [1] [2]

E. G. Williams, Fourier Acoustics, Academic Press, 1999. F. Deblauwe, J. Leuridan, J. L. Chauray, B. Béguet, "Acoustic holography in transient conditions", 6th Inter. Cong. Sound Vibration, Lyngby, Denmark, 5-8 July 1999, pp. 899-906. [3] J. Hald, "STSF - A Unique technique for scan-based Near-field Acoustic Holography without restrictions on coherence", Brüel & Kjaer Technical Review No. 1 (1989) 1-50. [4] J. Hald, "Time Domain Acoustical Holography and its applications", Sound and Vibration, February 2001, pp. 16-24. [5] J.-H. Thomas, J.-C. Pascal, V. Grulier, S. Paillasseur, J.-C. Le Roux, "Real-Time Nearfield Acoustic Holography (RT-NAH): a technique for time-continuous reconstruction of a source signal", NOVEM 2005, Saint-Raphaël, France, 18-21 April 2005. [6] E. G. Williams, "Regularization methods for near-field acoustical holography", J. Acoust. Soc. Am. 110 (4) (2001) 1976-1988. [7] V. Grulier, J.-H. Thomas, J.-C. Pascal, J.-C. Le Roux, "Time varying forward projection using wavenumber formulation", Inter-Noise 2004, Prague, Czech Republic, 22-25 August 2004. [8] V. Grulier, "Propagation directe et inverse dans l'espace temps-nombre d'onde : Application à une méthode d'holographie de champ proche", PhD dissertation, Université du Maine, 2005. [9] M. Forbes, S. Letcher, P. Stepanishen, "A wave vector, time-domain method of forward projecting time-dependent pressure fields", J. Acoust. Soc. Am. 90 (5) (1991) 2782-27 [10] S. Haykin, Adaptive Filter Theory, Prentice Hall, 1996.

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