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Spectral estimation method for noisy data using a noise reference
Applied Acoustics 72 (2011) 11–21
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Spectral estimation method for noisy data using a noise reference Daniel Blacodon ⇑ ONERA, BP 72, Châtillon 92322, France
a r t i c l e
i n f o
Article history: Received 7 December 2009 Received in revised form 15 June 2010 Accepted 14 September 2010
Keywords: Source localization Array processing Background noise
a b s t r a c t The development of array processing methods to extract the useful characteristics of acoustic sources such as their locations and absolute levels, starting from the measured sound field is one of the main issues in aero-acoustics. The conventional beamforming method is a very popular technique investigated to solve the power level estimation problem. It has the advantage of being robust, easy to implement and cheap in computation time. However, this technique is also known for having poor spatial resolution capabilities which prevents the correct source levels being obtained for numerous practical applications. Deconvolution techniques of the result computed with CBF, with the point spread function of the array manifold, may restore the power levels of the acoustic sources that would be observed in the absence of the array resolution effects. However, the accuracy of the results provided by deconvolution methods is very sensitive to background noise, always present in acoustic measurements. This process should be carried out after the additive noise has been suitably attenuated and, ideally, the deconvolution operator should amplify the noise as little as possible. Another approach is described in the article. It consists in using a noise reference and a new technique called spectral estimation method with additive noise to remove both the smearing effect produced by the array response and the background noise. The technique has been applied to computer and experimental simulations conducted both in an anechoic chamber and in the test section of an open wind tunnel involving acoustic sources radiating in a noisy environment. The levels of the sources were found with a good level of accuracy and the background noise greatly reduced, confirming the validity of the approach and the satisfactory performance of the method proposed. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Array processing is a powerful tool to extract noise source characteristics, both in terms of locations and levels in the frequency band of interest, from the acoustic signals measured during wind tunnel tests. For many years this means of investigation has been commonly used to study airframe noise [1–3], which is a significant contributor to the overall acoustic emissions of a civil aircraft when landing. The reduction of aerodynamic noise is a priority for the civilian aeronautics industry, which in the next decade will be confronted with more restrictive noise regulation in urban fly-over areas. In the classic multiple acoustic sources localization problem the sensor signals produced by the wave fields are processed in order to isolate the incident sources and estimate their associated positions. A wide variety of techniques have been developed to provide solutions to this problem. Conventional beamforming (CBF) is one of the most popular methods used as a diagnostic method because it supplies a source map in a single step at each computational ⇑ Tel.: +33 (1) 46 73 48 05. E-mail address: [email protected] 0003-682X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2010.09.004
frequency [4]. However, it should be emphasized that processing based on CBF is the optimum processing for a single monopole. The classic result is that the overall array response to a point source exhibits several sidelobes and a main lobe of which the location and amplitude correspond respectively to the location and amplitude of the point source. For multiple sources, the CBF method is able to perform the localization with a good degree of accuracy in many practical situations [5]. The power levels of the sources are accurately obtained when the sources are separated from each other by at least one 3 dB beam width, and when their amplitudes are larger than those of the side lobes. Unfortunately, the two previous criteria are not satisfied in situations of interest like aircraft noise, where the sources under study are continuously distributed, and the beam pattern associated with different sources overlap. In these cases, it is obvious that, because of its poor spatial resolution, the CBF method fails to provide correct levels. The estimation of the source power levels in a multiple source environment is essential in order to characterize spread sources like airframe noise. To solve the problem, we must resort to methods, which improve the overall performance of the CBF. An interesting solution has been suggested [6] for computing the total
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D. Blacodon / Applied Acoustics 72 (2011) 11–21
Nomenclature c R E[] f J L M Nq P0 Q t djk Ds
r2N s Xq(f)
340 m/s is speed of sound in the propagation medium distance between the source region and the array expectation operator frequency in Hertz number monopole sources to model D length of the array number of microphones of the array number of monopole sources in the source area q 20 lPa number of source areas in D time in seconds Kronecker delta minimum distance between two monopole sources variance of the white background noise time delay actual power spectral density of acoustic source area q
output of distributed sources. It consists in recovering the actual source power levels of specified areas from the CBF output, knowing the characteristics of the array pattern. The undesirable effects of smearing by the array pattern are removed from the output of the CBF by using a deconvolution technique that considers the beamform. However, the measured data not only contains the noise radiation from the source of interest, but also random noise, which always contaminates measurements. This situation arises, for example, in wind tunnels with a closed test section, where acoustic sources radiate in very a noisy environment, making it difficult to characterize them. Generally, the background noise has a spectrum, which is more extended than the spectra of the source areas, and it can often become dominant as the frequency increases. Consequently, the particular solutions associated with background noise may be superposed onto the solution of our problem, in an unfavourable way, by the normalization operation used to approximate the source power levels, leading to inaccurate results. The problem becomes critical when corrupting noise is changing the actual levels and the spectral distribution in the measured cross-spectral matrix. Besides, undesired dominant noise sources may be present in the corrupting noise, making it more difficult to decide in the localization maps whether the sources with high levels correspond to those of interest or not. In order to get over this drawback, a lot of research on the modeling and removal of the effects of noise and distortion has been done. The main conclusions of these studies are that the success of noise processing methods depends on their ability to characterize and model the noise process, and to use the noise characteristics advantageously to differentiate the signal from the noise [7]. Various techniques such as beamforming (non-adaptive and adaptive) and spatial–temporal filtering can be used to achieve noise reduction [8–10]. In certain applications, it is not possible to access the instantaneous frame of the contaminating noise, and only the noisy signal is available. Thus, the noise cannot be cancelled out, but it may be reduced, in an average sense, using the statistics of the signal and the noise process. In other situations, it is not possible to get a noise reference: the reference signal can either be generated artificially or extracted from the primary signal or from the imaging results [11]. Sometimes we have access to a noise reference and under some assumptions on the noise characteristics; several strategies exist that can be used to remove its effects. There is an interesting approach [12] that shows how the generalized singular value decomposition can be used to achieve noise reduction. Another popular solution uses the Spectral Subtraction method to reduce the noise
XCBF q ðf Þ
power spectral density of acoustic source area q estimated with CBF XSEM ðf Þ power spectral density of acoustic source area q estiq mated with SEM XSEMWAN ðf Þ power spectral density of acoustic source area q estiq mated with SEWMAN Abbreviations CBF conventional beamforming CSM cross-spectral matrix PSD power spectral density PSS power spectrum subtraction SEM spectral estimation method SEMWAN spectral estimation method with additive noise SPL spectral power level
contamination [13]. It is based on the subtraction of the short-term spectral magnitude of noise from that of the noisy spectrum. A generalized form of the basic spectral subtraction is given in [14] where power spectral subtraction uses short-time power spectrum estimates instead of a magnitude spectrum. Another interesting solution consists in removing the main diagonal of the array cross-spectral matrix before performing the imaging processing [15]. However, this solution is tractable when the background noise has both the same variance on all the microphones of the array and is also uncorrelated between the microphones. The present study is focused in the presentation on an extension of the spectral estimation method (SEM) which is based on a prior knowledge of a noise reference. This new method called spectral estimation method with additive noise (SEMWAN) has the advantage of being able to reduce the smearing effect due to the array response and at the same time the inaccuracy of the results caused by noise sources which can be coherent (i.e. the phase difference between the wave field emitted by the source is constant) as well as incoherent (i.e. the phase difference between the wave field radiated by the source is random), with high or low levels. This method is well suited to applications in a wind tunnel since we can, for example, get a noise reference or record the environmental noise before installing the models in the test section. This paper is organised as follows: Section 2 begins with a presentation of the problem we want to solve. Section 3 is devoted to a description of the array and model cross-spectral matrices used in the development of the methods considered in this study. A theoretical background for the estimation of the level of the acoustic sources with SEM and CBF, starting from noiseless data, is given in Section 4. The performances of CBF and SEM are compared in Section 5 using noiseless data. Section 6 presents the limitations of SEM when this method is used with noisy data. A short presentation of the Power Spectrum Subtraction is made in Section 7 and the principle of SEMWAN described in Section 8. SEM and SEMWAN are applied on numerical simulations, experimental simulations and in tests conducted in an open wind tunnel in Sections 9–11 respectively.
2. Problem of source levels estimation with noisy data Two extended sources S1 and S2 of power spectral densities (PSDs) X1(f) and X2(f) radiate in a very noisy environment. It is characterized by a coherent noise S3 of PSD X3(f), which can be between the acoustic sources or at the same location as one of these,
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D. Blacodon / Applied Acoustics 72 (2011) 11–21
Fig. 1. Geometry of the problem of acoustic SPLs estimation using noisy data.
and an incoherent broadband noise SN of variance r2N , as depicted in Fig. 1. The objective here is to find the location of S1 and S2 and estimate their absolute levels X1(f) and X2(f), starting from the noisy signals and the measurement of a noise reference. 3. Array and model cross-spectral matrices The methods studied in this paper are based on an array matrix and a model cross-spectral matrix and these are described below. 3.1. Array CSM
that an extended noise source may only be viewed as an equivalence class between source functions radiating the same pressure field on a phased array. A valuable equivalent source model could be a distribution of correlated monopoles, allowing the characterization of any directivity. However, the problem of finding both the amplitude of the sources and their correlation function in amplitude and phase involves a very large number of parameters, and it is not easy to solve with a limited number of microphones. In many applications, a majority of noise sources have smooth directivity patterns. This means that if the aperture angle of an array of microphones seen from the overall source region is not too large, the directivity pattern of each region may be considered as isotropic within this aperture. Therefore, a model based on a set of uncorrelated monopoles, like that used in the paper, is quite appropriate to study airframe noise. Let us consider that each monopole is characterized by a source function ai(t). The correlation function between two monopoles is:
C i;j ðsÞ ¼ E½ai ðtÞaj ðt þ sÞ ¼ C i ðsÞdi;j
The power spectral density of ai(t) is Ai(f), the Fourier Transform of Ci(s) (Eq. (3)). The problem of the source level estimation is then to find the J parameters Ai(f) of the model for each frequency. In order to proceed, it is necessary to define the general structure of the array CSM for uncorrelated monopole sources, and then choose the parameters Ai(f) such that the model CSM fits the SPLs measured as well as possible. To compute the model CSM, we begin by expressing the pressure radiated on the array:
pmod ðtÞ ¼ n
We consider an overall source domain D located in the (x, y) plane split into Q effective noise radiating areas Dq. An array of M microphones located in front of D records the radiated pressure. We assume that the acoustic sources are random and stationary. Let pmes n ðtÞ be the pressure recorded by the microphone n. The cross-correlation function C mes m;n ðsÞ between the microphone signals n and m is defined as: mes mes C mes m;n ðsÞ ¼ E½pm ðtÞpn ðt þ sÞ
ð1Þ
Cmes m;n ðf Þ ¼
Z
1
1
2pjf s C mes ds m;n ðsÞe
½C
mes
6 ðf Þ ¼ 4
C
mes 1;1 ðf Þ
C
mes 1;M ðf Þ
Cmes Cmes M;1 ðf Þ M;M ðf Þ
ð4Þ
Cmod m;n ðf Þ ¼
J X
Gm;i Ai ðf ÞGn;i
ð5Þ
i¼1 j2pfRmi c
with Gm;i ðf Þ ¼ e conjugation.
Rmi
, the superscript stands for the complex
ð2Þ
The measured array CSM of all signals pmes m ðtÞ (m = 1, 2, . . . , M) is defined by
2
J X ai ðt Rin =cÞ Rin i¼1
where Rin is the distance from the source i to the microphone n and J is the number of monopole sources considered for the description of all acoustic sub-regions. The model cross-power spectrum Cmod m;n ðf Þ, between the output signals of the mth and nth microphones, can be expressed as follows:
mes m;n ðf Þ
mes The cross-power spectral C between pmes m ðtÞ and pn ðtÞ is the Fourier Transform of C mes m;n ðsÞ (Wiener–Khintchine theorem):
ð3Þ
3 7 5
4. Source level estimation with noiseless data using SEM Consider ½Cmes ðf Þ, the array cross-spectral matrix, and ½Cmod ðf Þ, the model cross-spectral matrix due to the acoustic sources of component Cmod m;n ðf Þ (Eq. (5)). In SEM [5], the problem of finding the PSDs Ai(f) is based on the following cost function:
EðAðf ÞÞ ¼ 3.2. Model CSM In practical situations, it is not the details of noise sources such as the slats, the flaps and the landing gears for an aircraft at landing that are needed, but their directivity pattern in the far field to assess their contribution to the overall noise radiated. However, the problem is complex if we want to extract the directivity pattern of different extended source regions radiating simultaneously from the phased array data. At this point, the concept of extended noise source measured through its radiation must be clarified to get a better understanding of the true nature of the problem that we want to solve. It has been shown [16] that the measurement of the acoustic field outside the source region is not sufficient to determine the characteristics of the source uniquely. This implies
2 J M X X mes Gm;i ðf ÞAi ðf ÞGn;i ðf Þ Cm;n ðf Þ m;n¼1 i¼1
ð6Þ
The minimization of the mean squares error E(A(f)) between the array and model matrices with respect to Ai(f) gives the square linear system: J X
V i;j ðf ÞAi ðf Þ ¼ U j ðf Þ
ð7Þ
j¼1
P mes with U j ðf Þ ¼ M m;n¼1 Gm;j ðf ÞCm;n ðf ÞGm;j ðf Þ; 2 P V i;j ðf Þ ¼ Nm¼1 Gm;i ðf ÞGm;j ðf Þ . b i ðf Þ of the deThe solutions of Eq. (7) provide the estimates A sired PSDs Ai(f). However, the Ai(f) obtained after the minimization
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D. Blacodon / Applied Acoustics 72 (2011) 11–21
process are not guaranteed to be positive. This can lead to physically unrealistic solutions for the recovered spectra (i.e. spectra with negative amplitudes). Adding a positivity constraint on parameters Ai(f) can easily circumvent this drawback. b i ðf Þ depends on the variance of However, the accuracy of the A the background noise. So we consider that the acoustic sources radiate in an ideal medium without background noise. In this case, the array cross-spectral matrix [Cmes] may be assumed equal to [CS], the source cross-spectral matrix that would be obtained with noiseless data. Thus, Eq. (7) may be written as: J X
V i;j ðf ÞAi ðf Þ ¼ U Sj ðf Þ
ð8Þ
j¼1
with U Sj ðf Þ ¼
PM
m;n¼1 Gm;j ðf Þ
Cmod m;n ðf ÞGm;j ðf Þ.
is composed of 15 microphones equally spaced every Dx = 0. 1 cm: the length of the array is L = 1.4 m (Fig. 1).
5.1. Two sources spaced more than 3 dB beam width apart In the first test, S1 and S2 radiate at 4 kHz, so that their spatial separation DS = 0.2 m (i.e. DS = 2.35k, k = c/f = 0.085 m) is more than a standard beam width Dx3 dB ¼ 1:34k. The result found with CBF exhibits two main peaks (Fig. 2 upper plot). Their locations and amplitudes coincide with those of the actual sources S1 and S2. The spatial resolution of SEM is much better than that of the CBF (Fig. 2 lower plot) due to the spurious effect of the array response that has been removed from the result by solving the deconvolution
b i ðf Þ obtained by solving Eq. (8) will be used to The estimates A characterize the actual levels of the acoustic sources as it has been shown with numerical simulations [5]. 4.1. Source power estimation with SEM and CBF In this paper, we compare the results provided by SEM and CBF. It is thus necessary to define the total power level output from the source areas for both methods. 4.1.1. Source power estimation with SEM The total power level output XSEM ðf Þ of the source area Dq is obq b i ðf Þ of all monopole sources tained by summing the power levels A that are used in the modeling of sub region Dq:
XSEM q ðf Þ ¼
Nq X
b i ðf Þ A
ð9Þ
i¼1
The spectral estimation method (SEM) is the overall technique aiming to find XSEM ðf Þ. q 4.1.2. Source power levels estimation with CBF In the case of CBF, the estimation of the amplitude As(f) of a single compact source at location S is based on the following relation (more details are given in [4]): mes þ c ðf ; iÞ ¼ Gi ðf Þ½C ðf ÞGi ðf Þ W kGi ðf Þk4
Fig. 2. Scenario of two uncorrelated monopoles S1 ð70 dBÞ and S2 ð70 dBÞ spaced at more than one 3 dB bandwidth – results provided by the CFB method (upper plot) and SEM (lower plot).
ð10Þ
the vector Gþ i ðf Þ ¼ ½G1;i ðf Þ; G2;i ðf Þ; . . . ; GM;i ðf Þ and the superscript + stands for the transposed complex conjugate. It is pointed out here that the source map obtained by computing Eq. (10) over the source region Dq (for i = 1, . . . , Nq) will show a main lobe centered at the location of S and sidelobes due to the arc ðf ; iÞ is maximum at the location of S (i.e. for i = S) ray pattern. W c ðf ; iÞ ¼ As ðf Þ. and W The total power level output XCBF q ðf Þ of the source area Dq is obc ðf ; iÞ of all monopole tained by summing the power levels W sources that are used in the modeling of Dq:
XCBF q ðf Þ ¼
Nq X
c ðf ; iÞ W
ð11Þ
i¼1
5. Numerical simulations with noiseless data This section compares SEM and CBF using numerical simulations to show the capacity of both methods to estimate the actual levels of two compact sources S1 and S2 radiating in a noiseless environment. The sources are placed on a line of a length of 2 m, located at a distance R = 2 m. S1 is located at x = 0.1 m and S2 at x = +0.1. Both sources have the same amplitude of 70 dB. The array
Fig. 3. Scenario of two uncorrelated monopoles S1 ð70 dBÞ and S2 ð70 dBÞ spaced at less than one 3 dB bandwidth – results provided by the CFB method (upper plot) and SEM (lower plot).
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D. Blacodon / Applied Acoustics 72 (2011) 11–21
problem of Eq. (7). However, contrary to the CBF, the amplitudes of the two sources do not result from the maxima of the peaks. An integration over the spatial extended of the two peaks is needed to find the real levels of the sources. Indeed, the amplitudes of S1 and S2 are respectively provided with a good accuracy by computing XSEM ¼ 70:13 dB and XSEM ¼ 70:13 dB. 1 2 5.2. Two sources spaced less than 3 dB beam width apart The parameters of the simulation are the same as those used previously, except that here S1 and S2 radiate at 1.5 kHz, so are separated from each other by DS = 0.88k which is less than a standard beam width Dx3 dB ¼ 1:29k. CBF fails to resolve the two sources (Fig. 3 upper plot). However, the result exhibits a power level of 71.56 dB at the location of S1 and S2 (instead of 70 dB as expected). The source locations are correctly found with SEM (Fig. 3 lower plot). The integrated power levels XSEM ¼ 70:0 dB and XSEM ¼ 1 2 70:0 dB over the spatial extended of the two peaks correspond respectively at the actual power levels of S1 and S2. 6. Least squares solution for source level estimation with noisy data Now assuming that the wave fields radiated by the acoustic sources are contaminated by an additive noise, and that the source and the noise signals are uncorrelated. It follows that the true array cross-power spectral [Cmes,T(f)] obtained in practical cases can be expressed as the sum of [Cmes(f)] and [Cmes,N(f)] which are the cross-spectral matrices of the acoustic sources and the background noise respectively:
½Cmes;T ðf Þ ¼ ½Cmes ðf Þ þ ½Cmes;N ðf Þ
ð12Þ mes
It is again assumed that the cross-spectral matrix [C (f)] of the acoustic sources is equal to ½CS ðf Þ. Therefore the substitution of [Cmes(f)] with [Cmes,T(f)] in Eq. (7) yields: J X
V i;j ðf ÞAi ðf Þ ¼ U Sj ðf Þ þ U Nj ðf Þ
ð13Þ
i¼1
with U Nj ðf Þ ¼
PM
m;n¼1 Gm;j ðf Þ
Cmes;N m;n ðf ÞGm;j ðf Þ.
Eq. (13) may be written with the following compact form as: J X i¼1
" V i;j ðf ÞAi ðf Þ ¼ U Nj ðf Þ 1 þ
U Sj ðf Þ
# ð14Þ
U Nj ðf Þ
We may analyze Eq. (14) as follows: U S ðf Þ
(a) when the power ratio verifies U Nj ðf Þ 1, the estimated j parameters will allow an estimation of the acoustic source levels with a high degree of accuracy, U S ðf Þ
(b) when the power ratio verifies U Nj ðf Þ 1, the estimated j b i ðf Þ will characterize the undesired background parameters A noise, (c) In the other situations, actual PSDs of the acoustic source will be more or less corrupted by background noise.
because the computational complexity of the spectral subtraction is not expensive, it has a major drawback. In fact, the effectiveness of the noise removal process is dependent on obtaining an accurate spectral estimate of the noise signal. The better the noise estimate, the lower the level of residual noise appearing in the modified spectrum. However, since the noise spectrum cannot be directly obtained, it is necessary to use an averaged estimate of the noise. Hence, there are some significant variations between the estimated noise spectrum and the actual noise content present in the instantaneous spectra to denoise. The subtraction of these quantities results in the presence of isolated residual noise levels of large variance. The increase in the variance constitutes an irrevocable distortion in the reconstructed spectra.
8. Spectrum estimation method with additive noise The conclusion drawn in Section 6 is that SEM will give results that are strongly erroneous if it is applied to very noisy data. This drawback may be removed with the spectral estimation method with additive noise which is an improvement on SEM. This implies taking into account the contribution of the background noise in the acoustic measurements, and modifying the optimization problem using Eqs. (6) and (12), so that we obtain.
HðAðf ÞÞ ¼
mes;N jCmes;T m;n ðf Þ Cm;n ðf Þ
m;n¼1
J X
Gm;i ðf ÞAi ðf ÞGn;i ðf Þj2
ð15Þ
i¼1
b mes;N ðf Þ is generally not available The cross-spectral matrix ½ C because the measurements of acoustic sources and noise sources cannot be obtained separately. Nevertheless, in many practical situations we can measure the environmental noise with the array of microphones and thus access a reference noise. This data can be used to compute [Cmes,ref(N)(f)], called a reference noise cross-spectral matrix. At this point in the paper, we should consider several assumptions about the noise reference and the acoustic sources: (a) the noise reference is locally stationary, so that its magnitude measured before the tests without the acoustic sources is equal to its level produced during the tests with the acoustic sources, (b) the source signals and additive noise components are statistically independent. It follows from the above assumptions that [Cmes,ref(N)(f)] can be considered to be very good characterization of [Cmes,N(f)], so we can replace [Cmes,N(f)] with [Cmes,ref(N)(f)] in Eq. (15):
2 J M X X mes;T mes;refðNÞ HðAðf ÞÞ ¼ ðf Þ Gm;i ðf ÞAi ðf ÞGn;i ðf Þ Cm;n ðf Þ Cm;n m;n¼1 i¼1 ð16Þ Let us develop Eq. (16) in the following form:
HðAðf ÞÞ ¼ Bmes ðf Þ 2
J X
U mes;T ðf ÞAi ðf Þ þ 2 i
i¼1
7. Power spectrum subtraction The conclusions of the previous section show that the lower the magnitude of the background noise then the more accurate are the estimates of the sources PSDs. In other words, when noisy data is processed, a reduction of the noise will generally improve the accuracy of array processing methods based on the resolution of Eq. (13). A very simple way to achieve this, when a noise reference is available, consists in subtracting the noise reference from the noisy spectra. However, while this solution is very attractive
M X
þ BN ðf Þ 2Bmes;N ðf Þ þ
J X
Ai ðf ÞV i;j ðf ÞAj ðf Þ
With M X M X
2 mes;T jCmes;T ðf Þ m;n ðf Þj ; U i
m¼1 n¼1
¼
M X M X m¼1 n¼1
U Ni ðf ÞAi ðf Þ
i¼1
i;j
Bmes ðf Þ ¼
J X
Gm;i ðf ÞCmes;T m;n ðf ÞGn;i ðf Þ
ð17Þ
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D. Blacodon / Applied Acoustics 72 (2011) 11–21
U Ni ðf Þ ¼
M X M X
and from a numerical point of view [19]. An efficient welldebugged software version of the algorithm is available [20]. The minimization of the cost function H(a(f)) with respect to ai(f) provides positive noise-free PSDs as it will be shown in the next section. After this step, the total power level output XSEMWAN ðf Þ of q the source area Dq is obtained by summing the power levels a2i ðf Þ of all monopole sources that are used in the modeling of Dq:
mes;ref ðNÞ Gm;i ðf ÞCm;n ðf ÞGn;i ðf Þ; BN ðf Þ
m¼1 n¼1
¼
M X M X
ðNÞ jCmes;ref ðf Þj2 m;n
m¼1 n¼1
Bmes;N ðf Þ ¼
N X N X
ðNÞ Cmes;T ðf ÞCmes;ref ðf Þ and V i;j ðf Þ m;n m;n
m¼1 n¼1
XSEMWAN ðf Þ ¼ q
2 X M Gm;i ðf ÞGm;j ðf Þ ¼ m¼1
Nq X
a^ 2i ðf Þ
ð19Þ
i¼1
In this form, it appears that the cost function H(A(f)) (Eq. (17)) takes into account: (a) the bias due to the variance of the background noise with BN(f), (b) the background noise that was propagated in the source domain with U Ni ðf Þ, (c) the background that may be correlated with the acoustic sources with Bmes,N(f). The noise free power spectral densities Ai(f) are defined as the solution of the following problem:
b Þ ¼ arg min HðAðf ÞÞ Aðf
We will call this overall technique, to find XSEMWAN ðf Þ, the spectral q estimation method (SEMWAN). The result is valid only around the direction ~ u and in the aperture D~ u of the measuring array, seen from the source region. Consequently, in the acoustic far field, the SPL at a large distance R from the sources and around the direction ~ u is modeled as:
Pðf ; R~ uÞ ¼
Q X XSEMWAN ðf Þ q q¼1
R2
ð20Þ
In conclusion, XSEMWAN ðf Þ may be considered as the mean direcq tivity pattern of each radiating area q in the direction ~ u averaged on u. Then, rotating the measuring array around the global radiating D~ region enables us to estimate the full directivity pattern Pðf ; ~ uÞ.
A
As for SEM, the power spectral densities Ai obtained after the minimization process (Eq. (16)) are not guaranteed to be positive. This can lead to physically unrealistic solutions for the recovered spectra (i.e. spectra with negative amplitudes). In order to get over this drawback, a new set of parameters is introduced such as a2i ðf Þ ¼ Ai ðf Þ. Thus, the least-squares problem can be formulated as follows:
Hðaðf ÞÞ ¼ Bmes ðf Þ 2
J X
U mes;T ðf Þa2i ðf Þ þ 2 i
i¼1
þ BN ðf Þ 2Bmes;N ðf Þ þ
J X
U Ni ðf Þa2i ðf Þ
i¼1 J X
a2i ðf ÞV i;j ðf Þa2j ðf Þ
ð18Þ
i;j
9. Numerical simulations with noisy data The purpose of this section is to evaluate the capacity of SEMWAN to accurately estimate the levels and locations of two extended acoustic sources S1 and S2 radiating at a frequency of 4 kHz in a noisy environment (Fig. 4). S1 and S2 are composed of 40 uncorrelated monopoles evenly spaced with Ds = 0.005 m. The levels of each monopole source for S1 and S2 are 20 logðA1 Þ ¼ 55 dB and 20 logðA2 Þ ¼ 70 dB respectively, in decibels. The integrated power levels for S1 is given by the following relation:
X1 ¼ 10 log
40 X
A2i ¼ 71:02 dB
ð21Þ
i¼1
H(a(f)) is a polynomial of fourth degree, and its derivative a polynomial of third degree. Now, finally, two questions arise: on the existence of a unique global maximum (uniqueness of the solution) and on the choice of the initial guess to find a global minimum (and not a local one). It is generally difficult to answer these questions theoretically when the estimation problem has nonlinearity characteristics such as the one that is the subject of this paper. Today, we may only say that we have obtained different solutions and minima starting from very different values of the initial guess. However, it has been found in all the simulations performed that the initial guess a2i ðf Þ ¼ U mes;T ðf Þ (Eq. (18)) leads to the global minimum, although i we lack general proof for this observation. Furthermore, the minimization of the cost function H(a(f)) with respect to ai provides positive noise-free PSDs.
If we substitute A1 by A2 in Eq. (21), we find that the integrated power level for S2 is equal to X2 ¼ 86:02 dB. The wave fields generated by the two sources are assumed to be measured by the same array of microphones considered in Section 5. In order to simulate the noisy environment the virtual microphone signals are contaminated by an additive coherent noise source S3 composed of 24 uncorrelated monopole sources, each with a level of 20 logðA3 Þ ¼ 50 dB such that the integrated power levels X3 ¼ 63:8 dB, and also with an incoherent noise SN with a variance in decibels equal to ðr2B ÞdB ¼ 50 dB.
8.1. Source power levels computation The computation procedure consists first in sampling the source areas, then in determining the power levels a2i ðf Þ of virtual uncorrelated monopoles located at each of the sampling points, by minimizing the error in Eq. (18). From all the methods available to perform nonlinear optimization [17], we chose an efficient and robust algorithm called the Restarted Conjugate Gradient Method to compute the a2i ðf Þ. This method is well documented from a statistical point of view [18]
Fig. 4. Simulation of two extended sources S1 and S2 contaminated by a coherent noise source S3 and a broadband noise SN with a variance of 50 dB.
D. Blacodon / Applied Acoustics 72 (2011) 11–21
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9.1. Results with CBF–SEM and SEMWAN for the first scenario Fig. 5 shows the result given by conventional beamforming (CBF) and SEM at the emission frequency of 4 kHz for the configuration presented in Fig. 4. Predictably, CBF neither resolves the two extended sources S1 and S2 nor gives their actual levels. In contrast, SEM separates S1 and S2 and correctly characterizes the two sources in terms of both location and levels. It also appears that the coherent spurious source S3 is found with SEM. However, it will be difficult to decide in practical situations if S3 is a source of interest or not. We also observe several unwanted sources due to the incoherent background noise. Fig. 6a shows the noise reference (i.e. S3 and SN) used to compute [Cmes,ref(N)(f)] and in Fig. 6b the result provided by SEMWAN. Now, the spurious sources are removed and S1 and S2 are found at their actual locations with their levels within 1 dB.
Fig. 7. Simulation of two extended sources S1 and S2 contaminated by a coherent noise source S3 and a broadband noise SN with a variance of 100 dB.
9.2. Results with CBF–SEM and SEMWAN for the second scenario We consider in Fig. 7 a configuration with a high level of 100 dB for SN. This situation may be representative of the tests carried out in a wind tunnel with closed test section. The characteristics of the acoustic sources S1 and S2 remain unchanged compared to those presented in Fig. 4. In contrast, the spurious source S3 is now situated at the same location of S2 and is composed of 24 uncorrelated
Fig. 5. Result obtained with CBF and SEM for the configuration presented in Fig. 2.
Fig. 8. (a) Result obtained with CBF and SEM for the configuration presented in Fig. 7. (b) Result obtained with SEMWAN using the noise references S3 and SN.
monopole sources, each with a level of 20 logðA3 Þ ¼ 90 dB such that the integrated power levels X3 ¼ 103:8 dB. Again, CBF does not resolve S1 and S2 but it shows the location of the undesired source S3 (Fig. 8a). The degradation of the result found by SEM (Fig. 8a) is clear compared to that shown in Fig. 5. It is not possible to distinguish the sources S1 and S2 because the result is completely erratic over the whole computational domain. This is in agreement with the study described in Section 6. Fig. 8b shows the result obtained with SEMWAN. Now, S1 and S2 are resolved, their levels are correctly estimated and the smearing effect of the background noise removed. 10. Experimental simulations 10.1. Tests conducted in an anechoic chamber
Fig. 6. (a) Noise references used to compute [Cmes,ref(N)(f)] and apply SEMWAN for the configuration presented in Fig. 4. (b) Result obtained with SEMWAN.
A simple experiment was conducted in an anechoïc chamber to validate SEMWAN with experimental data. A photograph and a schematic view of the experimental set-up are presented in Fig. 9a and b respectively. We consider here that the acoustic sources are three driver units S1, S2 and S3 mounted in the middle of a wooden plate and spatially separated at 0.29 m. The background noise is generated by the driver units S4 and S5 placed near the array of microphones and between the two wooden plates 3 m
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from each other. The phased array is composed of 30 microphones, 1/2 in. 3211 Brüel&Kjaer, evenly spaced at 3 cm.
10.2. Methodology of the tests The driver units S1, S2 and S3 are the acoustic sources. They were connected independently at three generators delivering sinusoidal signals at the frequency of 2.5 kHz, 4.83 kHz and 8 kHz, respectively. The driver units S4 and S5 are the noise sources. They were also connected to independent generators delivering a broadband signal filtered at 10 kHz for S4 and a sinusoidal signal at 3 kHz for S5. The acoustic waves radiated by each of the five driver units were individually measured by the array of microphones. The spectra of S1, S2, S3, S4 and S5 are presented in the Fig. 10a–e respectively. The levels of the measured spectra by the microphones 29, 15 and 1 opposite the sources S1, S2, S3 respectively will be compared to the results provided by SEM and SEMWAN. Thus, it will possible to assess the capacity of both methods to accurately reproduce the levels of acoustic sources radiating in a noisy environment. The noise reference matrix [Cmes,ref(N)(f)] required by SEMWAN is computed with the data measured when only S4 and S5 impinge on the array of microphones. An example of the spectra of [Cmes,ref(N)(f)] is presented in Fig. 10f. It shows the broadband noise and tone noise emerging, of about 10 dB at 3 kHz.
10.3. Data processing
Fig. 9. (a) Photograph of the experimental set-up in an anechoic chamber used to validate SEMWAN. (b) Schematic view of the experiment set-up.
The array cross-matrices [Cmes,T(f)] and [Cmes,ref(N)(f)] were estimated using 200 data blocks, of 1024 samples each, sampled at 31,250 Hz (i.e. Df = 31,250/1024 = 30.51 Hz). The levels of
Fig. 10. Spectra of the acoustic sources: (a) Tone noise at 2.5 kHz. (b) Tone noise at 4.83 kHz. (c) Tone noise at 8 kHz. Spectra of the noise sources. (d) White noise filtered with a low pass filter at 10 kHz. (e) Tone noise at 3 kHz. (f) Spectrum of S4 + S5.
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Nq = 200 virtual monopole sources were computed at spatial sampling points equally spaced every Dy = 0.005 m, on a line passing through the middle of the driver units S1, S2 and S3, from y = 0.4 m to y = 1.4 m, at the discrete frequencies fg = gDf (g = 1, . . . , 328), and in the frequency range [0, 10 kHz]. The measured spectra are compared to the estimated spectra with SEM and SEMWAN after their propagation at the location of microphone n using the following relation:
PN q Pn ðf Þ ¼ 10 log
a^ 2i ðf Þ i¼1 R2 ni
P20
ð22Þ
10.4. Results This section presents a result obtained with Power Spectrum Subtraction (PSS), and with SEM and SEMWAN, firstly in the characterization of the sources S1 and S2, then S2 and S3 when the array measurements are contaminated by S4 and S5.
Fig. 12. (a) Result found with SEM for the configuration shown in Fig. 11a. (b) Result found with SEMWAN.
10.4.1. Result with PSS The Fig. 11a presents the spectra of S1 (solid line) and S2 (dashed line), and the spectrum of the signal Microphone 16 (black curve) when S1, S2, S4 and S5 impinge on the array. In Fig. 11b the spectrum of Microphone 16 (thick line) and the result found with PSS (grey curve) are superimposed. Reduction of the background is obtained with PSS, but there is a lot of frequency components with negative levels, certainly due to the weak signal-to-noise ratio considered in this test. We can also see that the sources S1 and S2 do not clearly appear at their emission frequencies. This result is a good illustration of the drawback of PSS. 10.4.2. Results with SEM and SEMWAN The same configuration is used as that considered in the previous section. Fig. 12a shows the result given by SEM. The level of S1 is correct, while that of S2 is underestimated by about 3 dB. There are also numerous spurious sources with high levels generated by the noise sources S4 and S5. The result given by SEMWAN is presented in Fig. 12b. The noise sources have been highly reduced and the levels of S1 and S2 found within 1 dB. This demonstrates the efficiency of SEMWAN for reducing the undesired noise sources when a reference of these noise sources is available. The second scenario is shown in Fig. 13a. The objective here is the characterization of S2 and S3 starting from noisy measurements of S4 and S5. The graph in Fig. 13b shows the result with SEM. The levels of source S2 is underestimated by about 3 dB while that of S3 is within 1.5 dB. Again, there are numerous spurious sources with
Fig. 13. (a) Spectra of acoustic sources S2 (solid line) and S3 (dashed line) to be estimated and the spectrum measured by microphone 16 (thick line) due to S2, S3, S4 and S5. (b) Result found with SEM. (c) Result found with SEMWAN.
high levels due to S4 and S5. The result obtained with SEMWAN (Fig. 13c) shows that S2 and S3 appear more clearly due to the reduction of noise. The level of S1 is accurately estimated and that of S2 obtained within 2.54 dB. It also appears that S5 was not completely removed by SEMWAN. 11. Tests conducted in an open wind tunnel
Fig. 11. (a) Comparison between the noisy spectrum on microphone 16 due to S1, S2, S4 and S5. (b) Result obtained with Power Spectrum Subtraction using the noise reference given in Fig. 10f.
Tests similar to those carried out in the anechoic chamber were conducted in the test section of the Onera open wind tunnel F2 (Fig. 14). The acoustic sources are three driver units S1, S2 and S3 installed in the ceiling of the test section. S1, S2 and S3 were
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Fig. 14. (a) Photograph of the experimental set-up in the Onera open wind tunnel F2. (b) Schematic view of the experiment set-up.
connecting to three independent generators delivering sinus signals of frequencies 2.5 kHz, 4.83 kHz and 8 kHz respectively. In this study, the noise sources are characterized by the noise radiated by the fan of the wind tunnel and by the turbulent boundary layer flowing over the cross-shaped array installed in the floor of the test section. During the tests, the noise radiated by the acoustic sources are measured using a cross-shaped array, equipped with 50 half-inch microphones AKSUD 3211 evenly spaced at 3 cm on each arm. Thirty microphones were distributed along the arm of the array, which is in the y-axis, parallel to the flow direction. In contrast, twenty microphones were distributed along the arm in the x-axis,
perpendicular to the flow direction. Because the antenna was located in the flow, a convection effect occurs between the acoustic sources and the locations of the microphones. This is taken into account by applying flow corrections on the propagation time [21,22] which are used in the computation of the model array CSM. The noisy cross-spectral matrix [Cmes,T(f)] and the noise reference matrix [Cmes,ref(N)(f)] needed to apply SEMWAN were estimated using 1800 data blocks, of 1024 samples each, sampled at a frequency of 62,500 Hz (i.e. Df = 62,500/1024 = 61.035 Hz). It should be pointed out here that [Cmes,ref(N)(f)] is computed with signals measured by the array of microphones when the three sinus generators connected to S1, S2 and S3 were off.
Fig. 15. Comparison of the spectra measured with microphone 15 (see Fig. 14) and those computed with SEM and SEMWAN when the flow is at speeds V = 10 m/s and 50 m/s. The spectra noise reference measured with microphone 15 are also shown for the two flow speeds.
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The levels of 2700 virtual monopole sources were computed at the nodes of a grid (from x = 2.06 to 2.95 m and from y = 0.15 m to 0.15 m) with an elementary mesh of size Ds = Dx Dy (with Dx = Dy = 0.01 m) at the discrete frequencies fg = gDf(g = 1, . . . , 250) in the frequency range [0, 15 kHz]. 11.1. Results The aim now is to assess the capability of SEM and SEMWAN to restore the frequency components of S1, S2 and S3 for two configurations when the data measured by the array is corrupted by low and high background noise. The overall measured spectrum with microphone 15 near the centre of the array (see Fig. 14) up to the frequency 15 kHz is compared to the integrated power levels computed with the SEM and SEMWAN. 11.1.1. Test at V = 10 m/s At this low speed of flow the background generated in the wind tunnel weakly corrupts the data measured by the array of microphones. It follows that the spectral components of S1, S2 and S3 (Fig. 15a) are correctly found with SEM. We also observe that the source levels are quite similar to those shown by the spectrum of microphone 15 (see Fig. 14). Fig. 15b exhibits the result obtained with SEMWAN using a cross-spectral matrix of the noise reference computed with data measured when S1, S2 and S3 did not impinge on the array. Clearly, the three acoustic sources are correctly characterized in frequency and the background noise level has been reduced for the whole of the frequency band [0, 15 kHz]. It is lower than the background noise measured with microphone 15 across almost all the frequency band considered in this test. However, the result provided by SEMWAN does not show more information than that provided by SEM. The results provided by SEMWAN does not show more information than the spectra provided by SEM. It is obvious that the radiation of the acoustic sources is not done in free field but in reverberant conditions. Therefore, it is not possible to conclude that levels of the sources found with SEMWAN are equal to the actual ones simply by comparing the estimated and measured spectra. 11.1.2. Test at V = 50 m/s At a speed of flow of 50 m/s the background noise measured with microphone 15 (Fig. 15a) has increased by about 25 dB compared to that measured at 10 m/s (Fig. 15a). SEM correctly restores the frequency components of S1 and S2 (Fig. 15c) which are at higher levels than the background noise. In contrast S3 does not appear at all due to its weak level. The reduction of 15 dB of the background obtained with SEMWAN means it is now possible to clearly show the three acoustic sources. However, as for the previous test, it is not possible to conclude that levels of the sources found with SEMWAN are equal to the actual ones simply by comparing the estimated and measured spectra. 12. Summary This study has presented a new method for estimating the real levels of acoustic sources radiated in a noisy environment. This method called spectral estimation method with additive noise (SEMWAN) is based on the prior knowledge of a noise reference. The minimization of a cost function including the noise reference cross-spectral matrix, the array and model matrices, gives
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solutions with highly reduced background noise. SEMWAN has been applied to computer, experimental simulations and tests conducted in an open wind tunnel to estimate the levels of acoustic sources radiated in a noisy environment. The results have confirmed the validity of the approach and the performance of the proposed method. In particular, the levels of the acoustic sources simulated numerically or tested in the anechoic chamber were restituted with a good degree of accuracy. Concerning the test conducted in the open wind tunnel, further investigations remain to be done in order to assess the influence of the reverberation effects on the source levels provided by SEMWAN. This will require examining techniques of de-reverberation that have been proposed [7]. Acknowledgements The author is grateful to MM. Williams Denis and fabrice Desmerger for many stimulating discussions. This work was supported by the Department of Numerical Simulation and Aeroacoustics at Onera as part of the MASA home project. References [1] Piet JF, Élias G. Airframe noise source localization using a microphone array. AIAA/CEAS aeroacoustics conference, 3rd, Atlanta, GA, 12–14 May, 1997. AIAA Paper No. 1997-1643. [2] Davy D, Remy H. Airframe noise characteristics of a 1/11 model airbus. AIAA/ CEAS aeroacoustics conference, 4th, Toulouse (France), 2–4 June, 1998. AIAA Paper No. 98-2335. [3] Blacodon D. Analysis of the airframe noise of an A320/A321 with a parametric method. J Aircraft 2007;44(1):26–34. [4] Élias G. Source localization with a two-dimensional focused array-optimal processing for a cross-shaped array. In: Inter-Noise 95, Newport Beach (USA), 10–12 July 1995. [5] Blacodon D, Elias G. Level estimation of extended acoustic sources using a parametric method. J Aircraft 2004;41(6):1360–9. [6] Brooks T. A deconvolution approach for the mapping of acoustic sources (DAMAS) determined from phased microphone arrays. In: 10th AIAA/CEAS aeroacoustics conference, Manchester, Great Britain, 10–12 May, 2004. AIAA Paper No. 2004-2954. [7] Benesty J, Chen J, Huang Y. Microphone array signal processing. Springer; 2008. [8] Cox H, Zeskind RM, Owen MM. Robust adaptive beamforming. In: IEEE trans. acoust., speech, signal process, vol. 35 (10).; 1987. p. 1365–75. [9] Capon J. High resolution frequency–wavenumber spectrum analysis. Proc IEEE 1969;57:1408–18. [10] Frost OL. An algorithm for linearly constrained adaptive array processing. Proc IEEE 1972;60:926–35. [11] Koop L. Microphone-array processing for wind-tunnel measurements with strong background noise. In: 14th AIAA/CEAS aeroacoustics conference, Vancouver, British Columbia, 5–7 May, 2008. AIAA Paper No. 2008-2907 [12] Bulté J. Acoustic array measurements in aerodynamic wind tunnels: a subspace approach for noise suppression. In: 13th AIAA/CEAS aeroacoustics conference, Rome, Italy, 21–23 May, 2007. AIAA Paper No. 2007-3446. [13] Boll SF. Suppression of acoustic noire in speech using spectral subtraction. IEEE Trans Acoustic Speech Signal Process 1979;27:113–20. [14] Berouti M, Schwartz- R, Makhoul J. Enhancement of speech corrupted by acoustic noise. Proc IEEE Int Conf Acoust 1979;4:208–11. [15] Sijtsma P. CLEAN based on spatial source coherence. In: 13th AIAA/CEAS aeroacoustics conference, Rome, Italy, 21–23 May, 2007. AIAA Paper No. 20073436. [16] Dowling AP, Ffowcs Williams JE. Sound and sources of sound. Chichester: Ellis Horwood Publishers; 1983. [17] Gill PE, Muttey W, Wright MH. Practical optimization. London: Academic Press, Inc.; 1981. p. 59–66. [18] Bard Y. Nonlinear parameter estimation. New York: Academic Press; 1974. [19] Bandler JW. Computed-aided circuit optimization. In: Temes GC, Mitra SK, editors. Modern filter theory and design. New York: Wiley; 1973. [20] Shanno DF, Phua KH. Minimization of unconstrained multivariate functions. ACM transactions on mathematical software 6; 1980. p. 618–22. [21] Candel S, Guédel A, Julienne A. Radiation, refraction and scattering on waves in a shear flow. AIAA Paper-76-544. American Institute of Aeronautics and Astronautics. Aero-acoustics conference, 3rd, Palo Alto, Calif., Jul. 20–22, 1976. [22] Amiet RK. Refraction of sound by a shear layer. J Sound Vib 1978;58(3):467–82.
Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Spectral estimation method for noisy data using a noise reference Daniel Blacodon ⇑ ONERA, BP 72, Châtillon 92322, France
a r t i c l e
i n f o
Article history: Received 7 December 2009 Received in revised form 15 June 2010 Accepted 14 September 2010
Keywords: Source localization Array processing Background noise
a b s t r a c t The development of array processing methods to extract the useful characteristics of acoustic sources such as their locations and absolute levels, starting from the measured sound field is one of the main issues in aero-acoustics. The conventional beamforming method is a very popular technique investigated to solve the power level estimation problem. It has the advantage of being robust, easy to implement and cheap in computation time. However, this technique is also known for having poor spatial resolution capabilities which prevents the correct source levels being obtained for numerous practical applications. Deconvolution techniques of the result computed with CBF, with the point spread function of the array manifold, may restore the power levels of the acoustic sources that would be observed in the absence of the array resolution effects. However, the accuracy of the results provided by deconvolution methods is very sensitive to background noise, always present in acoustic measurements. This process should be carried out after the additive noise has been suitably attenuated and, ideally, the deconvolution operator should amplify the noise as little as possible. Another approach is described in the article. It consists in using a noise reference and a new technique called spectral estimation method with additive noise to remove both the smearing effect produced by the array response and the background noise. The technique has been applied to computer and experimental simulations conducted both in an anechoic chamber and in the test section of an open wind tunnel involving acoustic sources radiating in a noisy environment. The levels of the sources were found with a good level of accuracy and the background noise greatly reduced, confirming the validity of the approach and the satisfactory performance of the method proposed. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Array processing is a powerful tool to extract noise source characteristics, both in terms of locations and levels in the frequency band of interest, from the acoustic signals measured during wind tunnel tests. For many years this means of investigation has been commonly used to study airframe noise [1–3], which is a significant contributor to the overall acoustic emissions of a civil aircraft when landing. The reduction of aerodynamic noise is a priority for the civilian aeronautics industry, which in the next decade will be confronted with more restrictive noise regulation in urban fly-over areas. In the classic multiple acoustic sources localization problem the sensor signals produced by the wave fields are processed in order to isolate the incident sources and estimate their associated positions. A wide variety of techniques have been developed to provide solutions to this problem. Conventional beamforming (CBF) is one of the most popular methods used as a diagnostic method because it supplies a source map in a single step at each computational ⇑ Tel.: +33 (1) 46 73 48 05. E-mail address: [email protected] 0003-682X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2010.09.004
frequency [4]. However, it should be emphasized that processing based on CBF is the optimum processing for a single monopole. The classic result is that the overall array response to a point source exhibits several sidelobes and a main lobe of which the location and amplitude correspond respectively to the location and amplitude of the point source. For multiple sources, the CBF method is able to perform the localization with a good degree of accuracy in many practical situations [5]. The power levels of the sources are accurately obtained when the sources are separated from each other by at least one 3 dB beam width, and when their amplitudes are larger than those of the side lobes. Unfortunately, the two previous criteria are not satisfied in situations of interest like aircraft noise, where the sources under study are continuously distributed, and the beam pattern associated with different sources overlap. In these cases, it is obvious that, because of its poor spatial resolution, the CBF method fails to provide correct levels. The estimation of the source power levels in a multiple source environment is essential in order to characterize spread sources like airframe noise. To solve the problem, we must resort to methods, which improve the overall performance of the CBF. An interesting solution has been suggested [6] for computing the total
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Nomenclature c R E[] f J L M Nq P0 Q t djk Ds
r2N s Xq(f)
340 m/s is speed of sound in the propagation medium distance between the source region and the array expectation operator frequency in Hertz number monopole sources to model D length of the array number of microphones of the array number of monopole sources in the source area q 20 lPa number of source areas in D time in seconds Kronecker delta minimum distance between two monopole sources variance of the white background noise time delay actual power spectral density of acoustic source area q
output of distributed sources. It consists in recovering the actual source power levels of specified areas from the CBF output, knowing the characteristics of the array pattern. The undesirable effects of smearing by the array pattern are removed from the output of the CBF by using a deconvolution technique that considers the beamform. However, the measured data not only contains the noise radiation from the source of interest, but also random noise, which always contaminates measurements. This situation arises, for example, in wind tunnels with a closed test section, where acoustic sources radiate in very a noisy environment, making it difficult to characterize them. Generally, the background noise has a spectrum, which is more extended than the spectra of the source areas, and it can often become dominant as the frequency increases. Consequently, the particular solutions associated with background noise may be superposed onto the solution of our problem, in an unfavourable way, by the normalization operation used to approximate the source power levels, leading to inaccurate results. The problem becomes critical when corrupting noise is changing the actual levels and the spectral distribution in the measured cross-spectral matrix. Besides, undesired dominant noise sources may be present in the corrupting noise, making it more difficult to decide in the localization maps whether the sources with high levels correspond to those of interest or not. In order to get over this drawback, a lot of research on the modeling and removal of the effects of noise and distortion has been done. The main conclusions of these studies are that the success of noise processing methods depends on their ability to characterize and model the noise process, and to use the noise characteristics advantageously to differentiate the signal from the noise [7]. Various techniques such as beamforming (non-adaptive and adaptive) and spatial–temporal filtering can be used to achieve noise reduction [8–10]. In certain applications, it is not possible to access the instantaneous frame of the contaminating noise, and only the noisy signal is available. Thus, the noise cannot be cancelled out, but it may be reduced, in an average sense, using the statistics of the signal and the noise process. In other situations, it is not possible to get a noise reference: the reference signal can either be generated artificially or extracted from the primary signal or from the imaging results [11]. Sometimes we have access to a noise reference and under some assumptions on the noise characteristics; several strategies exist that can be used to remove its effects. There is an interesting approach [12] that shows how the generalized singular value decomposition can be used to achieve noise reduction. Another popular solution uses the Spectral Subtraction method to reduce the noise
XCBF q ðf Þ
power spectral density of acoustic source area q estimated with CBF XSEM ðf Þ power spectral density of acoustic source area q estiq mated with SEM XSEMWAN ðf Þ power spectral density of acoustic source area q estiq mated with SEWMAN Abbreviations CBF conventional beamforming CSM cross-spectral matrix PSD power spectral density PSS power spectrum subtraction SEM spectral estimation method SEMWAN spectral estimation method with additive noise SPL spectral power level
contamination [13]. It is based on the subtraction of the short-term spectral magnitude of noise from that of the noisy spectrum. A generalized form of the basic spectral subtraction is given in [14] where power spectral subtraction uses short-time power spectrum estimates instead of a magnitude spectrum. Another interesting solution consists in removing the main diagonal of the array cross-spectral matrix before performing the imaging processing [15]. However, this solution is tractable when the background noise has both the same variance on all the microphones of the array and is also uncorrelated between the microphones. The present study is focused in the presentation on an extension of the spectral estimation method (SEM) which is based on a prior knowledge of a noise reference. This new method called spectral estimation method with additive noise (SEMWAN) has the advantage of being able to reduce the smearing effect due to the array response and at the same time the inaccuracy of the results caused by noise sources which can be coherent (i.e. the phase difference between the wave field emitted by the source is constant) as well as incoherent (i.e. the phase difference between the wave field radiated by the source is random), with high or low levels. This method is well suited to applications in a wind tunnel since we can, for example, get a noise reference or record the environmental noise before installing the models in the test section. This paper is organised as follows: Section 2 begins with a presentation of the problem we want to solve. Section 3 is devoted to a description of the array and model cross-spectral matrices used in the development of the methods considered in this study. A theoretical background for the estimation of the level of the acoustic sources with SEM and CBF, starting from noiseless data, is given in Section 4. The performances of CBF and SEM are compared in Section 5 using noiseless data. Section 6 presents the limitations of SEM when this method is used with noisy data. A short presentation of the Power Spectrum Subtraction is made in Section 7 and the principle of SEMWAN described in Section 8. SEM and SEMWAN are applied on numerical simulations, experimental simulations and in tests conducted in an open wind tunnel in Sections 9–11 respectively.
2. Problem of source levels estimation with noisy data Two extended sources S1 and S2 of power spectral densities (PSDs) X1(f) and X2(f) radiate in a very noisy environment. It is characterized by a coherent noise S3 of PSD X3(f), which can be between the acoustic sources or at the same location as one of these,
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Fig. 1. Geometry of the problem of acoustic SPLs estimation using noisy data.
and an incoherent broadband noise SN of variance r2N , as depicted in Fig. 1. The objective here is to find the location of S1 and S2 and estimate their absolute levels X1(f) and X2(f), starting from the noisy signals and the measurement of a noise reference. 3. Array and model cross-spectral matrices The methods studied in this paper are based on an array matrix and a model cross-spectral matrix and these are described below. 3.1. Array CSM
that an extended noise source may only be viewed as an equivalence class between source functions radiating the same pressure field on a phased array. A valuable equivalent source model could be a distribution of correlated monopoles, allowing the characterization of any directivity. However, the problem of finding both the amplitude of the sources and their correlation function in amplitude and phase involves a very large number of parameters, and it is not easy to solve with a limited number of microphones. In many applications, a majority of noise sources have smooth directivity patterns. This means that if the aperture angle of an array of microphones seen from the overall source region is not too large, the directivity pattern of each region may be considered as isotropic within this aperture. Therefore, a model based on a set of uncorrelated monopoles, like that used in the paper, is quite appropriate to study airframe noise. Let us consider that each monopole is characterized by a source function ai(t). The correlation function between two monopoles is:
C i;j ðsÞ ¼ E½ai ðtÞaj ðt þ sÞ ¼ C i ðsÞdi;j
The power spectral density of ai(t) is Ai(f), the Fourier Transform of Ci(s) (Eq. (3)). The problem of the source level estimation is then to find the J parameters Ai(f) of the model for each frequency. In order to proceed, it is necessary to define the general structure of the array CSM for uncorrelated monopole sources, and then choose the parameters Ai(f) such that the model CSM fits the SPLs measured as well as possible. To compute the model CSM, we begin by expressing the pressure radiated on the array:
pmod ðtÞ ¼ n
We consider an overall source domain D located in the (x, y) plane split into Q effective noise radiating areas Dq. An array of M microphones located in front of D records the radiated pressure. We assume that the acoustic sources are random and stationary. Let pmes n ðtÞ be the pressure recorded by the microphone n. The cross-correlation function C mes m;n ðsÞ between the microphone signals n and m is defined as: mes mes C mes m;n ðsÞ ¼ E½pm ðtÞpn ðt þ sÞ
ð1Þ
Cmes m;n ðf Þ ¼
Z
1
1
2pjf s C mes ds m;n ðsÞe
½C
mes
6 ðf Þ ¼ 4
C
mes 1;1 ðf Þ
C
mes 1;M ðf Þ
Cmes Cmes M;1 ðf Þ M;M ðf Þ
ð4Þ
Cmod m;n ðf Þ ¼
J X
Gm;i Ai ðf ÞGn;i
ð5Þ
i¼1 j2pfRmi c
with Gm;i ðf Þ ¼ e conjugation.
Rmi
, the superscript stands for the complex
ð2Þ
The measured array CSM of all signals pmes m ðtÞ (m = 1, 2, . . . , M) is defined by
2
J X ai ðt Rin =cÞ Rin i¼1
where Rin is the distance from the source i to the microphone n and J is the number of monopole sources considered for the description of all acoustic sub-regions. The model cross-power spectrum Cmod m;n ðf Þ, between the output signals of the mth and nth microphones, can be expressed as follows:
mes m;n ðf Þ
mes The cross-power spectral C between pmes m ðtÞ and pn ðtÞ is the Fourier Transform of C mes m;n ðsÞ (Wiener–Khintchine theorem):
ð3Þ
3 7 5
4. Source level estimation with noiseless data using SEM Consider ½Cmes ðf Þ, the array cross-spectral matrix, and ½Cmod ðf Þ, the model cross-spectral matrix due to the acoustic sources of component Cmod m;n ðf Þ (Eq. (5)). In SEM [5], the problem of finding the PSDs Ai(f) is based on the following cost function:
EðAðf ÞÞ ¼ 3.2. Model CSM In practical situations, it is not the details of noise sources such as the slats, the flaps and the landing gears for an aircraft at landing that are needed, but their directivity pattern in the far field to assess their contribution to the overall noise radiated. However, the problem is complex if we want to extract the directivity pattern of different extended source regions radiating simultaneously from the phased array data. At this point, the concept of extended noise source measured through its radiation must be clarified to get a better understanding of the true nature of the problem that we want to solve. It has been shown [16] that the measurement of the acoustic field outside the source region is not sufficient to determine the characteristics of the source uniquely. This implies
2 J M X X mes Gm;i ðf ÞAi ðf ÞGn;i ðf Þ Cm;n ðf Þ m;n¼1 i¼1
ð6Þ
The minimization of the mean squares error E(A(f)) between the array and model matrices with respect to Ai(f) gives the square linear system: J X
V i;j ðf ÞAi ðf Þ ¼ U j ðf Þ
ð7Þ
j¼1
P mes with U j ðf Þ ¼ M m;n¼1 Gm;j ðf ÞCm;n ðf ÞGm;j ðf Þ; 2 P V i;j ðf Þ ¼ Nm¼1 Gm;i ðf ÞGm;j ðf Þ . b i ðf Þ of the deThe solutions of Eq. (7) provide the estimates A sired PSDs Ai(f). However, the Ai(f) obtained after the minimization
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D. Blacodon / Applied Acoustics 72 (2011) 11–21
process are not guaranteed to be positive. This can lead to physically unrealistic solutions for the recovered spectra (i.e. spectra with negative amplitudes). Adding a positivity constraint on parameters Ai(f) can easily circumvent this drawback. b i ðf Þ depends on the variance of However, the accuracy of the A the background noise. So we consider that the acoustic sources radiate in an ideal medium without background noise. In this case, the array cross-spectral matrix [Cmes] may be assumed equal to [CS], the source cross-spectral matrix that would be obtained with noiseless data. Thus, Eq. (7) may be written as: J X
V i;j ðf ÞAi ðf Þ ¼ U Sj ðf Þ
ð8Þ
j¼1
with U Sj ðf Þ ¼
PM
m;n¼1 Gm;j ðf Þ
Cmod m;n ðf ÞGm;j ðf Þ.
is composed of 15 microphones equally spaced every Dx = 0. 1 cm: the length of the array is L = 1.4 m (Fig. 1).
5.1. Two sources spaced more than 3 dB beam width apart In the first test, S1 and S2 radiate at 4 kHz, so that their spatial separation DS = 0.2 m (i.e. DS = 2.35k, k = c/f = 0.085 m) is more than a standard beam width Dx3 dB ¼ 1:34k. The result found with CBF exhibits two main peaks (Fig. 2 upper plot). Their locations and amplitudes coincide with those of the actual sources S1 and S2. The spatial resolution of SEM is much better than that of the CBF (Fig. 2 lower plot) due to the spurious effect of the array response that has been removed from the result by solving the deconvolution
b i ðf Þ obtained by solving Eq. (8) will be used to The estimates A characterize the actual levels of the acoustic sources as it has been shown with numerical simulations [5]. 4.1. Source power estimation with SEM and CBF In this paper, we compare the results provided by SEM and CBF. It is thus necessary to define the total power level output from the source areas for both methods. 4.1.1. Source power estimation with SEM The total power level output XSEM ðf Þ of the source area Dq is obq b i ðf Þ of all monopole sources tained by summing the power levels A that are used in the modeling of sub region Dq:
XSEM q ðf Þ ¼
Nq X
b i ðf Þ A
ð9Þ
i¼1
The spectral estimation method (SEM) is the overall technique aiming to find XSEM ðf Þ. q 4.1.2. Source power levels estimation with CBF In the case of CBF, the estimation of the amplitude As(f) of a single compact source at location S is based on the following relation (more details are given in [4]): mes þ c ðf ; iÞ ¼ Gi ðf Þ½C ðf ÞGi ðf Þ W kGi ðf Þk4
Fig. 2. Scenario of two uncorrelated monopoles S1 ð70 dBÞ and S2 ð70 dBÞ spaced at more than one 3 dB bandwidth – results provided by the CFB method (upper plot) and SEM (lower plot).
ð10Þ
the vector Gþ i ðf Þ ¼ ½G1;i ðf Þ; G2;i ðf Þ; . . . ; GM;i ðf Þ and the superscript + stands for the transposed complex conjugate. It is pointed out here that the source map obtained by computing Eq. (10) over the source region Dq (for i = 1, . . . , Nq) will show a main lobe centered at the location of S and sidelobes due to the arc ðf ; iÞ is maximum at the location of S (i.e. for i = S) ray pattern. W c ðf ; iÞ ¼ As ðf Þ. and W The total power level output XCBF q ðf Þ of the source area Dq is obc ðf ; iÞ of all monopole tained by summing the power levels W sources that are used in the modeling of Dq:
XCBF q ðf Þ ¼
Nq X
c ðf ; iÞ W
ð11Þ
i¼1
5. Numerical simulations with noiseless data This section compares SEM and CBF using numerical simulations to show the capacity of both methods to estimate the actual levels of two compact sources S1 and S2 radiating in a noiseless environment. The sources are placed on a line of a length of 2 m, located at a distance R = 2 m. S1 is located at x = 0.1 m and S2 at x = +0.1. Both sources have the same amplitude of 70 dB. The array
Fig. 3. Scenario of two uncorrelated monopoles S1 ð70 dBÞ and S2 ð70 dBÞ spaced at less than one 3 dB bandwidth – results provided by the CFB method (upper plot) and SEM (lower plot).
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D. Blacodon / Applied Acoustics 72 (2011) 11–21
problem of Eq. (7). However, contrary to the CBF, the amplitudes of the two sources do not result from the maxima of the peaks. An integration over the spatial extended of the two peaks is needed to find the real levels of the sources. Indeed, the amplitudes of S1 and S2 are respectively provided with a good accuracy by computing XSEM ¼ 70:13 dB and XSEM ¼ 70:13 dB. 1 2 5.2. Two sources spaced less than 3 dB beam width apart The parameters of the simulation are the same as those used previously, except that here S1 and S2 radiate at 1.5 kHz, so are separated from each other by DS = 0.88k which is less than a standard beam width Dx3 dB ¼ 1:29k. CBF fails to resolve the two sources (Fig. 3 upper plot). However, the result exhibits a power level of 71.56 dB at the location of S1 and S2 (instead of 70 dB as expected). The source locations are correctly found with SEM (Fig. 3 lower plot). The integrated power levels XSEM ¼ 70:0 dB and XSEM ¼ 1 2 70:0 dB over the spatial extended of the two peaks correspond respectively at the actual power levels of S1 and S2. 6. Least squares solution for source level estimation with noisy data Now assuming that the wave fields radiated by the acoustic sources are contaminated by an additive noise, and that the source and the noise signals are uncorrelated. It follows that the true array cross-power spectral [Cmes,T(f)] obtained in practical cases can be expressed as the sum of [Cmes(f)] and [Cmes,N(f)] which are the cross-spectral matrices of the acoustic sources and the background noise respectively:
½Cmes;T ðf Þ ¼ ½Cmes ðf Þ þ ½Cmes;N ðf Þ
ð12Þ mes
It is again assumed that the cross-spectral matrix [C (f)] of the acoustic sources is equal to ½CS ðf Þ. Therefore the substitution of [Cmes(f)] with [Cmes,T(f)] in Eq. (7) yields: J X
V i;j ðf ÞAi ðf Þ ¼ U Sj ðf Þ þ U Nj ðf Þ
ð13Þ
i¼1
with U Nj ðf Þ ¼
PM
m;n¼1 Gm;j ðf Þ
Cmes;N m;n ðf ÞGm;j ðf Þ.
Eq. (13) may be written with the following compact form as: J X i¼1
" V i;j ðf ÞAi ðf Þ ¼ U Nj ðf Þ 1 þ
U Sj ðf Þ
# ð14Þ
U Nj ðf Þ
We may analyze Eq. (14) as follows: U S ðf Þ
(a) when the power ratio verifies U Nj ðf Þ 1, the estimated j parameters will allow an estimation of the acoustic source levels with a high degree of accuracy, U S ðf Þ
(b) when the power ratio verifies U Nj ðf Þ 1, the estimated j b i ðf Þ will characterize the undesired background parameters A noise, (c) In the other situations, actual PSDs of the acoustic source will be more or less corrupted by background noise.
because the computational complexity of the spectral subtraction is not expensive, it has a major drawback. In fact, the effectiveness of the noise removal process is dependent on obtaining an accurate spectral estimate of the noise signal. The better the noise estimate, the lower the level of residual noise appearing in the modified spectrum. However, since the noise spectrum cannot be directly obtained, it is necessary to use an averaged estimate of the noise. Hence, there are some significant variations between the estimated noise spectrum and the actual noise content present in the instantaneous spectra to denoise. The subtraction of these quantities results in the presence of isolated residual noise levels of large variance. The increase in the variance constitutes an irrevocable distortion in the reconstructed spectra.
8. Spectrum estimation method with additive noise The conclusion drawn in Section 6 is that SEM will give results that are strongly erroneous if it is applied to very noisy data. This drawback may be removed with the spectral estimation method with additive noise which is an improvement on SEM. This implies taking into account the contribution of the background noise in the acoustic measurements, and modifying the optimization problem using Eqs. (6) and (12), so that we obtain.
HðAðf ÞÞ ¼
mes;N jCmes;T m;n ðf Þ Cm;n ðf Þ
m;n¼1
J X
Gm;i ðf ÞAi ðf ÞGn;i ðf Þj2
ð15Þ
i¼1
b mes;N ðf Þ is generally not available The cross-spectral matrix ½ C because the measurements of acoustic sources and noise sources cannot be obtained separately. Nevertheless, in many practical situations we can measure the environmental noise with the array of microphones and thus access a reference noise. This data can be used to compute [Cmes,ref(N)(f)], called a reference noise cross-spectral matrix. At this point in the paper, we should consider several assumptions about the noise reference and the acoustic sources: (a) the noise reference is locally stationary, so that its magnitude measured before the tests without the acoustic sources is equal to its level produced during the tests with the acoustic sources, (b) the source signals and additive noise components are statistically independent. It follows from the above assumptions that [Cmes,ref(N)(f)] can be considered to be very good characterization of [Cmes,N(f)], so we can replace [Cmes,N(f)] with [Cmes,ref(N)(f)] in Eq. (15):
2 J M X X mes;T mes;refðNÞ HðAðf ÞÞ ¼ ðf Þ Gm;i ðf ÞAi ðf ÞGn;i ðf Þ Cm;n ðf Þ Cm;n m;n¼1 i¼1 ð16Þ Let us develop Eq. (16) in the following form:
HðAðf ÞÞ ¼ Bmes ðf Þ 2
J X
U mes;T ðf ÞAi ðf Þ þ 2 i
i¼1
7. Power spectrum subtraction The conclusions of the previous section show that the lower the magnitude of the background noise then the more accurate are the estimates of the sources PSDs. In other words, when noisy data is processed, a reduction of the noise will generally improve the accuracy of array processing methods based on the resolution of Eq. (13). A very simple way to achieve this, when a noise reference is available, consists in subtracting the noise reference from the noisy spectra. However, while this solution is very attractive
M X
þ BN ðf Þ 2Bmes;N ðf Þ þ
J X
Ai ðf ÞV i;j ðf ÞAj ðf Þ
With M X M X
2 mes;T jCmes;T ðf Þ m;n ðf Þj ; U i
m¼1 n¼1
¼
M X M X m¼1 n¼1
U Ni ðf ÞAi ðf Þ
i¼1
i;j
Bmes ðf Þ ¼
J X
Gm;i ðf ÞCmes;T m;n ðf ÞGn;i ðf Þ
ð17Þ
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D. Blacodon / Applied Acoustics 72 (2011) 11–21
U Ni ðf Þ ¼
M X M X
and from a numerical point of view [19]. An efficient welldebugged software version of the algorithm is available [20]. The minimization of the cost function H(a(f)) with respect to ai(f) provides positive noise-free PSDs as it will be shown in the next section. After this step, the total power level output XSEMWAN ðf Þ of q the source area Dq is obtained by summing the power levels a2i ðf Þ of all monopole sources that are used in the modeling of Dq:
mes;ref ðNÞ Gm;i ðf ÞCm;n ðf ÞGn;i ðf Þ; BN ðf Þ
m¼1 n¼1
¼
M X M X
ðNÞ jCmes;ref ðf Þj2 m;n
m¼1 n¼1
Bmes;N ðf Þ ¼
N X N X
ðNÞ Cmes;T ðf ÞCmes;ref ðf Þ and V i;j ðf Þ m;n m;n
m¼1 n¼1
XSEMWAN ðf Þ ¼ q
2 X M Gm;i ðf ÞGm;j ðf Þ ¼ m¼1
Nq X
a^ 2i ðf Þ
ð19Þ
i¼1
In this form, it appears that the cost function H(A(f)) (Eq. (17)) takes into account: (a) the bias due to the variance of the background noise with BN(f), (b) the background noise that was propagated in the source domain with U Ni ðf Þ, (c) the background that may be correlated with the acoustic sources with Bmes,N(f). The noise free power spectral densities Ai(f) are defined as the solution of the following problem:
b Þ ¼ arg min HðAðf ÞÞ Aðf
We will call this overall technique, to find XSEMWAN ðf Þ, the spectral q estimation method (SEMWAN). The result is valid only around the direction ~ u and in the aperture D~ u of the measuring array, seen from the source region. Consequently, in the acoustic far field, the SPL at a large distance R from the sources and around the direction ~ u is modeled as:
Pðf ; R~ uÞ ¼
Q X XSEMWAN ðf Þ q q¼1
R2
ð20Þ
In conclusion, XSEMWAN ðf Þ may be considered as the mean direcq tivity pattern of each radiating area q in the direction ~ u averaged on u. Then, rotating the measuring array around the global radiating D~ region enables us to estimate the full directivity pattern Pðf ; ~ uÞ.
A
As for SEM, the power spectral densities Ai obtained after the minimization process (Eq. (16)) are not guaranteed to be positive. This can lead to physically unrealistic solutions for the recovered spectra (i.e. spectra with negative amplitudes). In order to get over this drawback, a new set of parameters is introduced such as a2i ðf Þ ¼ Ai ðf Þ. Thus, the least-squares problem can be formulated as follows:
Hðaðf ÞÞ ¼ Bmes ðf Þ 2
J X
U mes;T ðf Þa2i ðf Þ þ 2 i
i¼1
þ BN ðf Þ 2Bmes;N ðf Þ þ
J X
U Ni ðf Þa2i ðf Þ
i¼1 J X
a2i ðf ÞV i;j ðf Þa2j ðf Þ
ð18Þ
i;j
9. Numerical simulations with noisy data The purpose of this section is to evaluate the capacity of SEMWAN to accurately estimate the levels and locations of two extended acoustic sources S1 and S2 radiating at a frequency of 4 kHz in a noisy environment (Fig. 4). S1 and S2 are composed of 40 uncorrelated monopoles evenly spaced with Ds = 0.005 m. The levels of each monopole source for S1 and S2 are 20 logðA1 Þ ¼ 55 dB and 20 logðA2 Þ ¼ 70 dB respectively, in decibels. The integrated power levels for S1 is given by the following relation:
X1 ¼ 10 log
40 X
A2i ¼ 71:02 dB
ð21Þ
i¼1
H(a(f)) is a polynomial of fourth degree, and its derivative a polynomial of third degree. Now, finally, two questions arise: on the existence of a unique global maximum (uniqueness of the solution) and on the choice of the initial guess to find a global minimum (and not a local one). It is generally difficult to answer these questions theoretically when the estimation problem has nonlinearity characteristics such as the one that is the subject of this paper. Today, we may only say that we have obtained different solutions and minima starting from very different values of the initial guess. However, it has been found in all the simulations performed that the initial guess a2i ðf Þ ¼ U mes;T ðf Þ (Eq. (18)) leads to the global minimum, although i we lack general proof for this observation. Furthermore, the minimization of the cost function H(a(f)) with respect to ai provides positive noise-free PSDs.
If we substitute A1 by A2 in Eq. (21), we find that the integrated power level for S2 is equal to X2 ¼ 86:02 dB. The wave fields generated by the two sources are assumed to be measured by the same array of microphones considered in Section 5. In order to simulate the noisy environment the virtual microphone signals are contaminated by an additive coherent noise source S3 composed of 24 uncorrelated monopole sources, each with a level of 20 logðA3 Þ ¼ 50 dB such that the integrated power levels X3 ¼ 63:8 dB, and also with an incoherent noise SN with a variance in decibels equal to ðr2B ÞdB ¼ 50 dB.
8.1. Source power levels computation The computation procedure consists first in sampling the source areas, then in determining the power levels a2i ðf Þ of virtual uncorrelated monopoles located at each of the sampling points, by minimizing the error in Eq. (18). From all the methods available to perform nonlinear optimization [17], we chose an efficient and robust algorithm called the Restarted Conjugate Gradient Method to compute the a2i ðf Þ. This method is well documented from a statistical point of view [18]
Fig. 4. Simulation of two extended sources S1 and S2 contaminated by a coherent noise source S3 and a broadband noise SN with a variance of 50 dB.
D. Blacodon / Applied Acoustics 72 (2011) 11–21
17
9.1. Results with CBF–SEM and SEMWAN for the first scenario Fig. 5 shows the result given by conventional beamforming (CBF) and SEM at the emission frequency of 4 kHz for the configuration presented in Fig. 4. Predictably, CBF neither resolves the two extended sources S1 and S2 nor gives their actual levels. In contrast, SEM separates S1 and S2 and correctly characterizes the two sources in terms of both location and levels. It also appears that the coherent spurious source S3 is found with SEM. However, it will be difficult to decide in practical situations if S3 is a source of interest or not. We also observe several unwanted sources due to the incoherent background noise. Fig. 6a shows the noise reference (i.e. S3 and SN) used to compute [Cmes,ref(N)(f)] and in Fig. 6b the result provided by SEMWAN. Now, the spurious sources are removed and S1 and S2 are found at their actual locations with their levels within 1 dB.
Fig. 7. Simulation of two extended sources S1 and S2 contaminated by a coherent noise source S3 and a broadband noise SN with a variance of 100 dB.
9.2. Results with CBF–SEM and SEMWAN for the second scenario We consider in Fig. 7 a configuration with a high level of 100 dB for SN. This situation may be representative of the tests carried out in a wind tunnel with closed test section. The characteristics of the acoustic sources S1 and S2 remain unchanged compared to those presented in Fig. 4. In contrast, the spurious source S3 is now situated at the same location of S2 and is composed of 24 uncorrelated
Fig. 5. Result obtained with CBF and SEM for the configuration presented in Fig. 2.
Fig. 8. (a) Result obtained with CBF and SEM for the configuration presented in Fig. 7. (b) Result obtained with SEMWAN using the noise references S3 and SN.
monopole sources, each with a level of 20 logðA3 Þ ¼ 90 dB such that the integrated power levels X3 ¼ 103:8 dB. Again, CBF does not resolve S1 and S2 but it shows the location of the undesired source S3 (Fig. 8a). The degradation of the result found by SEM (Fig. 8a) is clear compared to that shown in Fig. 5. It is not possible to distinguish the sources S1 and S2 because the result is completely erratic over the whole computational domain. This is in agreement with the study described in Section 6. Fig. 8b shows the result obtained with SEMWAN. Now, S1 and S2 are resolved, their levels are correctly estimated and the smearing effect of the background noise removed. 10. Experimental simulations 10.1. Tests conducted in an anechoic chamber
Fig. 6. (a) Noise references used to compute [Cmes,ref(N)(f)] and apply SEMWAN for the configuration presented in Fig. 4. (b) Result obtained with SEMWAN.
A simple experiment was conducted in an anechoïc chamber to validate SEMWAN with experimental data. A photograph and a schematic view of the experimental set-up are presented in Fig. 9a and b respectively. We consider here that the acoustic sources are three driver units S1, S2 and S3 mounted in the middle of a wooden plate and spatially separated at 0.29 m. The background noise is generated by the driver units S4 and S5 placed near the array of microphones and between the two wooden plates 3 m
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D. Blacodon / Applied Acoustics 72 (2011) 11–21
from each other. The phased array is composed of 30 microphones, 1/2 in. 3211 Brüel&Kjaer, evenly spaced at 3 cm.
10.2. Methodology of the tests The driver units S1, S2 and S3 are the acoustic sources. They were connected independently at three generators delivering sinusoidal signals at the frequency of 2.5 kHz, 4.83 kHz and 8 kHz, respectively. The driver units S4 and S5 are the noise sources. They were also connected to independent generators delivering a broadband signal filtered at 10 kHz for S4 and a sinusoidal signal at 3 kHz for S5. The acoustic waves radiated by each of the five driver units were individually measured by the array of microphones. The spectra of S1, S2, S3, S4 and S5 are presented in the Fig. 10a–e respectively. The levels of the measured spectra by the microphones 29, 15 and 1 opposite the sources S1, S2, S3 respectively will be compared to the results provided by SEM and SEMWAN. Thus, it will possible to assess the capacity of both methods to accurately reproduce the levels of acoustic sources radiating in a noisy environment. The noise reference matrix [Cmes,ref(N)(f)] required by SEMWAN is computed with the data measured when only S4 and S5 impinge on the array of microphones. An example of the spectra of [Cmes,ref(N)(f)] is presented in Fig. 10f. It shows the broadband noise and tone noise emerging, of about 10 dB at 3 kHz.
10.3. Data processing
Fig. 9. (a) Photograph of the experimental set-up in an anechoic chamber used to validate SEMWAN. (b) Schematic view of the experiment set-up.
The array cross-matrices [Cmes,T(f)] and [Cmes,ref(N)(f)] were estimated using 200 data blocks, of 1024 samples each, sampled at 31,250 Hz (i.e. Df = 31,250/1024 = 30.51 Hz). The levels of
Fig. 10. Spectra of the acoustic sources: (a) Tone noise at 2.5 kHz. (b) Tone noise at 4.83 kHz. (c) Tone noise at 8 kHz. Spectra of the noise sources. (d) White noise filtered with a low pass filter at 10 kHz. (e) Tone noise at 3 kHz. (f) Spectrum of S4 + S5.
D. Blacodon / Applied Acoustics 72 (2011) 11–21
19
Nq = 200 virtual monopole sources were computed at spatial sampling points equally spaced every Dy = 0.005 m, on a line passing through the middle of the driver units S1, S2 and S3, from y = 0.4 m to y = 1.4 m, at the discrete frequencies fg = gDf (g = 1, . . . , 328), and in the frequency range [0, 10 kHz]. The measured spectra are compared to the estimated spectra with SEM and SEMWAN after their propagation at the location of microphone n using the following relation:
PN q Pn ðf Þ ¼ 10 log
a^ 2i ðf Þ i¼1 R2 ni
P20
ð22Þ
10.4. Results This section presents a result obtained with Power Spectrum Subtraction (PSS), and with SEM and SEMWAN, firstly in the characterization of the sources S1 and S2, then S2 and S3 when the array measurements are contaminated by S4 and S5.
Fig. 12. (a) Result found with SEM for the configuration shown in Fig. 11a. (b) Result found with SEMWAN.
10.4.1. Result with PSS The Fig. 11a presents the spectra of S1 (solid line) and S2 (dashed line), and the spectrum of the signal Microphone 16 (black curve) when S1, S2, S4 and S5 impinge on the array. In Fig. 11b the spectrum of Microphone 16 (thick line) and the result found with PSS (grey curve) are superimposed. Reduction of the background is obtained with PSS, but there is a lot of frequency components with negative levels, certainly due to the weak signal-to-noise ratio considered in this test. We can also see that the sources S1 and S2 do not clearly appear at their emission frequencies. This result is a good illustration of the drawback of PSS. 10.4.2. Results with SEM and SEMWAN The same configuration is used as that considered in the previous section. Fig. 12a shows the result given by SEM. The level of S1 is correct, while that of S2 is underestimated by about 3 dB. There are also numerous spurious sources with high levels generated by the noise sources S4 and S5. The result given by SEMWAN is presented in Fig. 12b. The noise sources have been highly reduced and the levels of S1 and S2 found within 1 dB. This demonstrates the efficiency of SEMWAN for reducing the undesired noise sources when a reference of these noise sources is available. The second scenario is shown in Fig. 13a. The objective here is the characterization of S2 and S3 starting from noisy measurements of S4 and S5. The graph in Fig. 13b shows the result with SEM. The levels of source S2 is underestimated by about 3 dB while that of S3 is within 1.5 dB. Again, there are numerous spurious sources with
Fig. 13. (a) Spectra of acoustic sources S2 (solid line) and S3 (dashed line) to be estimated and the spectrum measured by microphone 16 (thick line) due to S2, S3, S4 and S5. (b) Result found with SEM. (c) Result found with SEMWAN.
high levels due to S4 and S5. The result obtained with SEMWAN (Fig. 13c) shows that S2 and S3 appear more clearly due to the reduction of noise. The level of S1 is accurately estimated and that of S2 obtained within 2.54 dB. It also appears that S5 was not completely removed by SEMWAN. 11. Tests conducted in an open wind tunnel
Fig. 11. (a) Comparison between the noisy spectrum on microphone 16 due to S1, S2, S4 and S5. (b) Result obtained with Power Spectrum Subtraction using the noise reference given in Fig. 10f.
Tests similar to those carried out in the anechoic chamber were conducted in the test section of the Onera open wind tunnel F2 (Fig. 14). The acoustic sources are three driver units S1, S2 and S3 installed in the ceiling of the test section. S1, S2 and S3 were
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D. Blacodon / Applied Acoustics 72 (2011) 11–21
Fig. 14. (a) Photograph of the experimental set-up in the Onera open wind tunnel F2. (b) Schematic view of the experiment set-up.
connecting to three independent generators delivering sinus signals of frequencies 2.5 kHz, 4.83 kHz and 8 kHz respectively. In this study, the noise sources are characterized by the noise radiated by the fan of the wind tunnel and by the turbulent boundary layer flowing over the cross-shaped array installed in the floor of the test section. During the tests, the noise radiated by the acoustic sources are measured using a cross-shaped array, equipped with 50 half-inch microphones AKSUD 3211 evenly spaced at 3 cm on each arm. Thirty microphones were distributed along the arm of the array, which is in the y-axis, parallel to the flow direction. In contrast, twenty microphones were distributed along the arm in the x-axis,
perpendicular to the flow direction. Because the antenna was located in the flow, a convection effect occurs between the acoustic sources and the locations of the microphones. This is taken into account by applying flow corrections on the propagation time [21,22] which are used in the computation of the model array CSM. The noisy cross-spectral matrix [Cmes,T(f)] and the noise reference matrix [Cmes,ref(N)(f)] needed to apply SEMWAN were estimated using 1800 data blocks, of 1024 samples each, sampled at a frequency of 62,500 Hz (i.e. Df = 62,500/1024 = 61.035 Hz). It should be pointed out here that [Cmes,ref(N)(f)] is computed with signals measured by the array of microphones when the three sinus generators connected to S1, S2 and S3 were off.
Fig. 15. Comparison of the spectra measured with microphone 15 (see Fig. 14) and those computed with SEM and SEMWAN when the flow is at speeds V = 10 m/s and 50 m/s. The spectra noise reference measured with microphone 15 are also shown for the two flow speeds.
D. Blacodon / Applied Acoustics 72 (2011) 11–21
The levels of 2700 virtual monopole sources were computed at the nodes of a grid (from x = 2.06 to 2.95 m and from y = 0.15 m to 0.15 m) with an elementary mesh of size Ds = Dx Dy (with Dx = Dy = 0.01 m) at the discrete frequencies fg = gDf(g = 1, . . . , 250) in the frequency range [0, 15 kHz]. 11.1. Results The aim now is to assess the capability of SEM and SEMWAN to restore the frequency components of S1, S2 and S3 for two configurations when the data measured by the array is corrupted by low and high background noise. The overall measured spectrum with microphone 15 near the centre of the array (see Fig. 14) up to the frequency 15 kHz is compared to the integrated power levels computed with the SEM and SEMWAN. 11.1.1. Test at V = 10 m/s At this low speed of flow the background generated in the wind tunnel weakly corrupts the data measured by the array of microphones. It follows that the spectral components of S1, S2 and S3 (Fig. 15a) are correctly found with SEM. We also observe that the source levels are quite similar to those shown by the spectrum of microphone 15 (see Fig. 14). Fig. 15b exhibits the result obtained with SEMWAN using a cross-spectral matrix of the noise reference computed with data measured when S1, S2 and S3 did not impinge on the array. Clearly, the three acoustic sources are correctly characterized in frequency and the background noise level has been reduced for the whole of the frequency band [0, 15 kHz]. It is lower than the background noise measured with microphone 15 across almost all the frequency band considered in this test. However, the result provided by SEMWAN does not show more information than that provided by SEM. The results provided by SEMWAN does not show more information than the spectra provided by SEM. It is obvious that the radiation of the acoustic sources is not done in free field but in reverberant conditions. Therefore, it is not possible to conclude that levels of the sources found with SEMWAN are equal to the actual ones simply by comparing the estimated and measured spectra. 11.1.2. Test at V = 50 m/s At a speed of flow of 50 m/s the background noise measured with microphone 15 (Fig. 15a) has increased by about 25 dB compared to that measured at 10 m/s (Fig. 15a). SEM correctly restores the frequency components of S1 and S2 (Fig. 15c) which are at higher levels than the background noise. In contrast S3 does not appear at all due to its weak level. The reduction of 15 dB of the background obtained with SEMWAN means it is now possible to clearly show the three acoustic sources. However, as for the previous test, it is not possible to conclude that levels of the sources found with SEMWAN are equal to the actual ones simply by comparing the estimated and measured spectra. 12. Summary This study has presented a new method for estimating the real levels of acoustic sources radiated in a noisy environment. This method called spectral estimation method with additive noise (SEMWAN) is based on the prior knowledge of a noise reference. The minimization of a cost function including the noise reference cross-spectral matrix, the array and model matrices, gives
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solutions with highly reduced background noise. SEMWAN has been applied to computer, experimental simulations and tests conducted in an open wind tunnel to estimate the levels of acoustic sources radiated in a noisy environment. The results have confirmed the validity of the approach and the performance of the proposed method. In particular, the levels of the acoustic sources simulated numerically or tested in the anechoic chamber were restituted with a good degree of accuracy. Concerning the test conducted in the open wind tunnel, further investigations remain to be done in order to assess the influence of the reverberation effects on the source levels provided by SEMWAN. This will require examining techniques of de-reverberation that have been proposed [7]. Acknowledgements The author is grateful to MM. Williams Denis and fabrice Desmerger for many stimulating discussions. This work was supported by the Department of Numerical Simulation and Aeroacoustics at Onera as part of the MASA home project. References [1] Piet JF, Élias G. Airframe noise source localization using a microphone array. AIAA/CEAS aeroacoustics conference, 3rd, Atlanta, GA, 12–14 May, 1997. AIAA Paper No. 1997-1643. [2] Davy D, Remy H. Airframe noise characteristics of a 1/11 model airbus. AIAA/ CEAS aeroacoustics conference, 4th, Toulouse (France), 2–4 June, 1998. AIAA Paper No. 98-2335. [3] Blacodon D. Analysis of the airframe noise of an A320/A321 with a parametric method. J Aircraft 2007;44(1):26–34. [4] Élias G. Source localization with a two-dimensional focused array-optimal processing for a cross-shaped array. In: Inter-Noise 95, Newport Beach (USA), 10–12 July 1995. [5] Blacodon D, Elias G. Level estimation of extended acoustic sources using a parametric method. J Aircraft 2004;41(6):1360–9. [6] Brooks T. A deconvolution approach for the mapping of acoustic sources (DAMAS) determined from phased microphone arrays. In: 10th AIAA/CEAS aeroacoustics conference, Manchester, Great Britain, 10–12 May, 2004. AIAA Paper No. 2004-2954. [7] Benesty J, Chen J, Huang Y. Microphone array signal processing. Springer; 2008. [8] Cox H, Zeskind RM, Owen MM. Robust adaptive beamforming. In: IEEE trans. acoust., speech, signal process, vol. 35 (10).; 1987. p. 1365–75. [9] Capon J. High resolution frequency–wavenumber spectrum analysis. Proc IEEE 1969;57:1408–18. [10] Frost OL. An algorithm for linearly constrained adaptive array processing. Proc IEEE 1972;60:926–35. [11] Koop L. Microphone-array processing for wind-tunnel measurements with strong background noise. In: 14th AIAA/CEAS aeroacoustics conference, Vancouver, British Columbia, 5–7 May, 2008. AIAA Paper No. 2008-2907 [12] Bulté J. Acoustic array measurements in aerodynamic wind tunnels: a subspace approach for noise suppression. In: 13th AIAA/CEAS aeroacoustics conference, Rome, Italy, 21–23 May, 2007. AIAA Paper No. 2007-3446. [13] Boll SF. Suppression of acoustic noire in speech using spectral subtraction. IEEE Trans Acoustic Speech Signal Process 1979;27:113–20. [14] Berouti M, Schwartz- R, Makhoul J. Enhancement of speech corrupted by acoustic noise. Proc IEEE Int Conf Acoust 1979;4:208–11. [15] Sijtsma P. CLEAN based on spatial source coherence. In: 13th AIAA/CEAS aeroacoustics conference, Rome, Italy, 21–23 May, 2007. AIAA Paper No. 20073436. [16] Dowling AP, Ffowcs Williams JE. Sound and sources of sound. Chichester: Ellis Horwood Publishers; 1983. [17] Gill PE, Muttey W, Wright MH. Practical optimization. London: Academic Press, Inc.; 1981. p. 59–66. [18] Bard Y. Nonlinear parameter estimation. New York: Academic Press; 1974. [19] Bandler JW. Computed-aided circuit optimization. In: Temes GC, Mitra SK, editors. Modern filter theory and design. New York: Wiley; 1973. [20] Shanno DF, Phua KH. Minimization of unconstrained multivariate functions. ACM transactions on mathematical software 6; 1980. p. 618–22. [21] Candel S, Guédel A, Julienne A. Radiation, refraction and scattering on waves in a shear flow. AIAA Paper-76-544. American Institute of Aeronautics and Astronautics. Aero-acoustics conference, 3rd, Palo Alto, Calif., Jul. 20–22, 1976. [22] Amiet RK. Refraction of sound by a shear layer. J Sound Vib 1978;58(3):467–82.