Real-time algorithm for acoustic imaging with a microphone array

Xun Huang: JASA Express Letters

关DOI: 10.1121/1.3100641兴

Published Online 9 April 2009

Real-time algorithm for acoustic imaging with a microphone array Xun Huang Department of Mechanical and Aerospace Engineering, College of Engineering, Peking University, Beijing, 100871, China [email protected]

Abstract: Acoustic phased array has become an important testing tool in aeroacoustic research, where the conventional beamforming algorithm has been adopted as a classical processing technique. The computation however has to be performed off-line due to the expensive cost. An innovative algorithm with real-time capability is proposed in this work. The algorithm is similar to a classical observer in the time domain while extended for the array processing to the frequency domain. The observer-based algorithm is beneficial mainly for its capability of operating over sampling blocks recursively. The expensive experimental time can therefore be reduced extensively since any defect in a testing can be corrected instantaneously. © 2009 Acoustical Society of America PACS numbers: 43.60.Fg, 43.60.Lq, 43.60.Mn [JC] Date Received: January 27, 2009 Date Accepted: February 19, 2009

1. Introduction Acoustic phased array1 has become an important tool in wind tunnel tests2 to identify the main noise source of an aircraft model3,4 with the objective of developing a quieter design5 that can reduce environmental impact. The conventional beamforming algorithm6 performed in the frequency domain1 was normally adopted as the fundamental processing method. The more advanced techniques, which include multiple signal classification7 and robust adaptive beamforming,8 worked unsatisfactorily in a noisy wind tunnel.9 The conventional beamforming algorithm has to be operated off-line due to the expensive cost in the computation of cross power matrix, which was computed by averaging over many sampling blocks. To achieve real-time computation a new recursive algorithm is presented in this paper. The derivation of the algorithm is based on the classical state observer in linear control theory.10 The observer-based algorithm performed in real-time will be quite helpful for wind tunnel tests since any defect in a test setup and experiments can be found and restored instantaneously. The expensive experimental time in a wind tunnel can therefore be reduced extensively. The rest of this paper is organized as follows. Section 2 briefly introduces the conventional beamforming algorithm in wind tunnel tests. The classical observer method is summarized in Sec. 3, followed by the development of the real-time algorithm that works recursively over each block of noisy wind tunnel signals. Section 4 presents a summary of this work. 2. Fundamentals of beamforming The beamforming algorithm for aeroacoustic testing is proceeded by firstly sampling the timedomain signal of acoustic pressure ym共t兲 from the mth microphone of a phased array through a data acquisition system. The sampling frequency is typically 40 kHz or higher. The time series of the digital samples is subsequently cut into blocks. The discrete Fourier transform (DFT) is performed over each block of ym共t兲 to produce the counterpart in the frequency domain, that is, 兩Ym共␻兲兩k for kth block at the angular frequency of ␻. The size of each block is typically a multiple of 4096 for the efficient computation of the DFT. In case of no confusion ␻ will be omitted in all following equations for conciseness. For a single noise source X at the position of ␰, the measurement satisfies

EL190 J. Acoust. Soc. Am. 125 共5兲, May 2009

© 2009 Acoustical Society of America

关DOI: 10.1121/1.3100641兴

Xun Huang: JASA Express Letters

Published Online 9 April 2009

0.4

0 -1 -2 -3 -4 -6 -7 -8 -9 -10

Y

0.2

0

-0.2

Background noise Dipole source -0.4

-0.4

-0.2

X

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Fig. 1. 共Color online兲 An acoustic source in a noisy testing facility.

兩Y兩k = 兩GX兩k ,

共1兲

where Y = 关Y1 , . . . , Yn兴⬘, G steering vector matrix, G = 关G1 , . . . , Gn兴⬘ 苸 Cn⫻1, the symbol of ⬘ denotes transpose, Gm is the steering vector that depends on the relative position between the noise source X and the mth microphone,5 and the value of n is the overall number of microphones in the phased array. In this work n = 56. The energy spectral density 共E兲 of X can be approximated by the conventional beamforming5 E = G*CG/储G储4 ,

共2兲

where the symbol of * denotes complex transpose, C is the cross-spectral matrix, C = 具兩Y兩k兩Y*兩k典, and 具 典 means the operation of averaging over a number of blocks. The sound pressure level (SPL) of the noise source X at the position of ␰ in decibels is P = 10 log10共E / 共4 ⫻ 10−10兲兲. An acoustic image for an aircraft model can thereafter be produced by performing the beamforming method over every ␰ of an area. One of the salient features in aerospace tests is the severe background noise from a wind tunnel facility. Figure 1 is an acoustic image of P at 5 kHz. The image includes a background noise that was measured from a tunnel with an open test section. The value of P is nondimensionalized to the maximal SPL in the domain. The rectangle in Fig. 1 represents the exit of the tunnel nozzle, where severe background noise can be found. The conventional beamforming algorithm in aeroacoustic research follows Welch’s method11 and can be proceeded in the following steps to remedy the effect of background noise. Step 1. A measurement is performed at the intended flow speed without the installation of aircraft model. The Fourier transformed outcome YB represents the sole result from the background noise. Step 2. A measurement is performed again at the same flow speed after an aircraft model is installed. The Fourier transformed outcome YBS is the collective results from the acoustic source of the model and the background noise. Step 3. Compute the cross-spectral matrix for YB and YBS, respectively, producing CB and CBS. With the assumption of little coherence12 between the acoustic source of the model and the background noise, the cross-spectral matrix for the model acoustic source can be approximated by YS = YBS − YB. As a result the interference from the background noise source XB can be minimized. Equation (2) is thereafter employed to obtain the amplitude of the acoustic source X S.

J. Acoust. Soc. Am. 125 共5兲, May 2009

Xun Huang: Real-time algorithm for acoustic imaging EL191

关DOI: 10.1121/1.3100641兴

Xun Huang: JASA Express Letters

Published Online 9 April 2009

0.4

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Y

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-0.4

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Fig. 2. 共Color online兲 Acoustic image by using the beamforming algorithm.

Step 4. Operate Eq. (2) continuously over a plane to find XS at different positions ␰ to generate an acoustic image. A simulated dipole source produced from an analytical solution was used as a benchmark in this work for the testing of the algorithms. The dipole source is superimposed onto the background noise which was measured in a wind tunnel. Figure 1 shows the dipole source that represents typical acoustic sources from an aircraft model, along with the severe background noise mainly from a noisy fan that produces the testing flow of a free stream at 30 m / s. Figure 2 is the acoustic imaging results obtained by performing the beamforming algorithm over K = 100 blocks. It can be seen that the interference from the background noise (mainly from the fan) is taken off satisfactorily. The beamforming algorithm is not real-time since the computation of the cross-spectral matrix has to be operated over many blocks. Excluding the cost of DFT, the rest cost of matrix multiplications over K blocks from total n microphones at single ␰ and single ␻ is proportional to Kn2. 3. Innovative real-time algorithm An innovative algorithm approximating the SPL of an acoustic source recursively is proposed in this work to achieve real-time performance. The origin of the algorithm is from the method of state observer in the linear control theory.10 It is worth mentioning that the original observer was presented for signals in the time domain.10 The idea was modified in this work to approximate the energy spectral density from data in the frequency domain. As a result, the observer-based algorithm proposed below is different from the existing recursive algorithms of adaptive beamforming that were normally operated in the time domain.13 From the perspective of the linear system theory, the state equation and the measurement equation of the acoustic testing in the frequency domain can be written as

兩X兩k+1 = 兩AX兩k ,

共3兲

兩Y兩k = 兩GX兩k .

共4兲

Subsequently a new matrix can be defined as

EL192 J. Acoust. Soc. Am. 125 共5兲, May 2009

Xun Huang: Real-time algorithm for acoustic imaging

关DOI: 10.1121/1.3100641兴

Xun Huang: JASA Express Letters

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(a)

Published Online 9 April 2009

-0.2

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(b)

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Fig. 3. 共Color online兲 Acoustic image by using state observer, where the results are from 共a兲 兩XˆBS兩k and 共b兲 兩XˆS兩k, k = 10.

冤 冥 G

O=

GA

] GAn−1

,

共5兲

where A is the state matrix that equals an identity matrix as long as the measured signal is stationary. The rest of the variables have been defined previously. As the rank of O equals n, all states X in Eq. (3) are observable10 from the microphone measurements. An observer can consequently be designed to approximate X from the measurements Y. The observer10 has the form of ˆ 兩 兲, 兩Xˆ兩k+1 = 兩AXˆ兩k + L共兩Y兩k − 兩Y k

共6兲

ˆ 兩 = 兩GXˆ兩 , 兩Y k k

共7兲

where the symbol of ˆ denotes the estimation of the states by the observer, and L is the observer gain. By subtracting Eq. (6) to Eq. (3), the error between the observation and the real signal at block k, 兩e兩k+1 = 兩Xˆ兩k+1 − 兩X兩k+1, satisfies 兩e兩k+1 = 共A − LG兲兩e兩k .

共8兲

As a result, the error e converges to zero when k → ⬁, as far as all eigenvalues of the matrix 共A − LG兲 are within a unit circle.10 Normally proper eigenvalues are firstly assigned and the observer gain L is solved accordingly. The computation of L can be performed off-line and the results can be stored in a table for the real-time computation of Eqs. (6) and (7). Figure 3(a) shows the result of 兩XˆBS兩k, which is obtained by performing Eqs. (6) and (7) recursively from the first block to the tenth block of the Fourier transformed outcome, YBS. In this work the eigenvalues of 共A − LG兲 were set to −0.5, for instance. Figure 3(a) suggests that the observer has the ability to reconstruct the dipole source. However, the acoustic image is still interfered by the background noise. To eliminate the effect of the background noise, the original state equations [Eqs. (3) and (4)] are modified to describe background noise and the acoustic source, respectively. The new state equations are

J. Acoust. Soc. Am. 125 共5兲, May 2009

Xun Huang: Real-time algorithm for acoustic imaging EL193

关DOI: 10.1121/1.3100641兴

Xun Huang: JASA Express Letters

Published Online 9 April 2009

冋 册 冋 册冋 册 冋 册 冋 册冋 册 兩XB兩k+1 兩XS兩k+1

兩YB兩k

兩YBS兩k

=

兩XB兩k

A

=

A

兩XB兩k

G

Ge

i␾

G

共9兲

,

兩XS兩k

兩XS兩k

共10兲

.

Basically two sets of experiments are required to obtain YB and YBS. The first experiment measured the background noise when an aircraft model is not installed in the test section. The second experiments measured the background noise plus the acoustic noise of the model. The symbol of ␾ denotes the phase difference between the background noise in the two experiments. The exact value of ␾ can be estimated by placing an extra sensor beside the dominant generator of the background noise. The corresponding equations of the state observer are

冋 册 冋 册冋 册 冉冋 册 冋 册冋 册冊 冋 册 冋 册冋 册
兩 兩X B k+1
兩 兩X S k+1

=


兩 兩X Bk

A

A


兩 兩X Sk


兩 兩Y Bk


兩 兩Y BS k

兩YB兩k

+L

=

兩YBS兩k

Ge

G


兩 兩X Bk

G

Ge


兩 兩X Bk

G

i␾

−


兩 兩X Sk

i␾

.

G


兩 兩X Sk

,

共11兲

共12兲

The computational cost of Eqs. (11) and (12) is proportional to n2 for each block. After finishing the acquisition of one block of data, Eqs. (11) and (12) can be applied to the block. In the meantime of the computation, the data acquisition system starts to obtain the next block of data. Although its overall cost is comparable to the cost of the conventional beamforming, the observer-based algorithm can be performed recursively in real-time. Figure 3(b) shows the outcome of acoustic image at tenth block. Compared to Fig. 3(a), the interference from the background noise is minimized clearly. The outcome also suggests a satisfactory convergence rate. 4. Summary A new algorithm with the real-time capability was presented for phased microphone arrays in this work. The algorithm was proposed from the perspective of the linear system theory and is similar to a classical observer in the form. It is worthwhile mentioning that the idealized assumptions such as free space of sound propagation and little sensor noise were hold for both the conventional beamforming and the observer-based algorithm. The method of Kalman filter, which can be regarded as an extended observer with the constraint of Gaussian noise, can be used to include practical imperfections such as multi-path, reflection, and sensor noise in the state model. As an extension of the present work the related research is ongoing and beyond the scope of this paper. In summary the present observer-based algorithm is able to be performed recursively over each sampling block. The outcome is still comparable to the corresponding beamforming result. The finding from the numerical experiment confirmed the working of the observer-based algorithm. Therefore, the expensive experimental time in a wind tunnel could be reduced extensively since any defect in a testing can be revealed and corrected instantaneously. The proposed algorithm should also be applicable to other areas, such as communications and ultrasonics. References and links 1

D. E. Dudgeon, “Fundamentals of digital array processing,” Proc. IEEE 65, 898–904 (1977). H.-C. Shin, W. R. Graham, P. Sijtsma, C. Andreou, and A. C. Faszer, “Implementation of a phased microphone array in a closed-section wind tunnel,” AIAA J. 45, 2897–2909 (2007). 3 Y. W. Wang, J. Li, P. Stoica, M. Sheplak, and T. Nishida, “Wideband relax and wideband clean for aeroacoustic imaging,” J. Acoust. Soc. Am. 115, 757–767 (2004). 4 Z. S. Wang, J. Li, P. Stoica, T. Nishida, and M. Sheplak, “Constant-beamwidth and constant-powerwidth wideband robust capon beamformers for acoustic imaging,” J. Acoust. Soc. Am. 116, 1621–1631 (2004). 2

EL194 J. Acoust. Soc. Am. 125 共5兲, May 2009

Xun Huang: Real-time algorithm for acoustic imaging

Xun Huang: JASA Express Letters

关DOI: 10.1121/1.3100641兴

Published Online 9 April 2009

5

T. J. E. Mueller, Aeroacoustic Measurements (Springer, Germany, 2002). B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE ASSP Mag. 5, 4–24 (1988). 7 R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag. 34, 276–280 (1986). 8 J. Li and P. Stoica, Robust Adaptive Beamforming (Wiley-Interscience, New York, 2005). 9 K. D. Donohue, J. Hannemann, and H. G. Dietz, “Performance of phase transform for detecting sound sources with microphone arrays in reverberant and noisy environments,” Signal Process. 87, 1677–1691 (2007). 10 S. Eduardo, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed. (Springer, Germany, 1998). 11 P. D. Welch, “The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967). 12 D. R. Morgan and T. M. Smith, “Coherence effects on the detection performance of quadratic array processors, with applications to large-array matched-field beamforming,” J. Acoust. Soc. Am. 87, 737–747 (1990). 13 S. N. Jagadeesha, S. N. Sinha, and D. K. Mehra, “A recursive modified Gram-Schmidt algorithm based adaptive beamformer,” Signal Process. 39, 69–78 (1994). 6

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Xun Huang: Real-time algorithm for acoustic imaging EL195