Comparative study on mode-identification algorithms using a phased-array system in a rectangular duct

Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Comparative study on mode-identification algorithms using a phased-array system in a rectangular duct Takao Suzuki n, Benjamin J. Day Acoustics and Fluid Mechanics, P.O. Box 3707, MC 0R-JF, The Boeing Company, Seattle, WA 98124-2207, United States

a r t i c l e i n f o

abstract

Article history: Received 30 October 2012 Received in revised form 12 June 2013 Accepted 21 June 2013 Handling Editor: K. Shin

To identify multiple acoustic duct modes, conventional beam-forming, CLEAN as well as L2 (i.e. pseudo-inverse) and L1 generalized-inverse beam-forming are applied to phased-array pressure data. A tone signal of a prescribed mode or broadband signal is generated upstream of a curved rectangular duct, and acoustic fields formed in both upstream and downstream stations of the test section are measured with identical wall-mounted microphone arrays. Sound-power distributions of several horizontal and vertical modes including upstream- and downstreampropagating waves can be identified with phased-array techniques, and the results are compared among the four approaches. The comparisons using synthetic data demonstrate that the L2 generalized-inverse algorithm can sufficiently suppress undesirable noise levels and detect amplitude distributions accurately in over-determined cases (i.e. the number of microphones is more than the number of cut-on modes) with minimum computational cost. As the number of cut-on modes exceeds the number of microphones (i.e. under-determined problems), the L1 algorithm is necessary to retain better accuracy. The comparison using test data acquired in the curved duct test rig (CDTR) at NASA Langley Research Center suggests that the L1 =L2 generalized-inverse approach as well as CLEAN can improve the dynamic range of the detected mode by as much as 10 dB relative to conventional beam-forming even with mean flow of M¼ 0.5. & 2013 Published by Elsevier Ltd.

1. Introduction Mode-identification techniques in duct acoustics are useful for aircraft-engine diagnosis and acoustic lining design. For example, intensities of discrete spinning modes from a fan, a compressor and/or a turbine in a turbo-fan engine can be inferred from phased-array pressure data acquired by sensors (e.g. microphones or Kulite sensors) in a duct. Compared with typical noise-source mapping, such as airframe-noise-source or jet-noise-source localization, modes to be detected in a duct are relatively well defined or even explicitly formulated, while coherency between the modes can be complicated. Higherorder spinning modes and scattered sound can be correlated with the fundamental tone, while sound from other sources may be incoherent. In many duct-acoustic applications, we may want to identify the intensity of each mode associated with a certain driving signal/motion in low signal-to-noise ratio environment. In the past, several mode-identification algorithms have been proposed and examined for duct acoustics. Primitive methods have been developed based on simple Fourier-decomposition [1,2] and advanced to pseudo-inverse approaches [3–6] for tonal noise or conventional beam-forming types [7,8] (i.e. phase-matching) for broadband noise. However, both pseudo-inverse and

n

Corresponding author. Tel.: +1 425 237 2921. E-mail address: [email protected] (T. Suzuki).

0022-460X/$ - see front matter & 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.jsv.2013.06.027

Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

2

conventional beam-forming approaches were developed based on least-squares optimization; consequently, the resolution of the mode distribution can be severely deteriorated due to contamination between mutual modes for certain conditions, resulting in unsatisfactory quantification (i.e. small dynamic range). A few efforts [9,10] have been made to improve the dynamic range for ductmode identification, for example by using CLEAN-SC [11]. While the primary mode can be distinctively extracted by this approach, other associated modes, such as higher-order spinning modes and scattered sound, are all removed. Moreover, these different approaches have not been compared in the past. Our particular interests here are cases in which the driving signal that excites modes to be detected is known and recordable. In engine tests, for example, the shaft rotational speeds are recorded by a tachometer simultaneously with signals from other sensors as time series; accordingly, a temporal wave-form (i.e. magnitude and phase) correlated with the shaft rotation can be extracted via the Fourier transforms synchronized with the tachometer signal. Through this process, the major part of uncorrelated noise can be filtered out. In typical noise-source mapping, a cross-spectral matrix is generated, and uncorrelated noise is suppressed by removing either lower-order eigenmodes or the diagonal part of the cross-spectral matrix [12]. When the driving mode is recordable, however, Fourier-transformed signals in a vector form can be directly processed. As a result, the noise level can be reduced and computational cost can be saved. In this paper, four different algorithms, conventional beam-forming, CLEAN [13], pseudo-inverse (i.e. L2) and L1 generalized-inverse beam-forming [14], typically used for noise-source mapping are formulated to process the aforementioned extracted signal in a vector form and examined to identify acoustic duct modes. In particular, an L1 generalizedinverse algorithm has been previously applied to beam-forming for a free jet [15], an impinging jet [16], flow past square cylinders [17] as well as airframe noise [14], and its superiority over various existing algorithms has been evaluated [14,16]. This algorithm is able to reconstruct multiple coherent modes and to detect accurate amplitudes in the presence of undesirable noise. By processing phased-array pressure data acquired from wall-mounted microphones in the curved duct test rig (CDTR) [18–20], we compare the mode distribution detected by L1 generalized-inverse beam-forming with those processed with the other algorithms. For both tonal and broadband signals, the L2 =L1 generalized-inverse methods as well as CLEAN provide better dynamic range (as much as 10 dB) than conventional beam-forming even with mean flow of M ¼0.5 in the duct. Moreover, we discuss advantages of the generalized-inverse algorithms over CLEAN depending on over- or underdetermined conditions in terms of robustness and computational cost. After the introduction, we explain the test facility briefly and then formulate the beam-forming algorithms tested in this study. Subsequently, we discuss the results of synthetic-data analysis and the test-data analysis in CDTR followed by the conclusions. 2. Description of the test facility An acoustic test was conducted at the CDTR facility in NASA Langley Research Center to evaluate acoustic lining attenuation. The test section, whose rectangular cross section measures 0.381 m  0.152 m (15 in  6 in), can be configured as a straight duct or as a curved duct smoothly connected to offset the exit one duct height (0.152 m ¼ 6 in) from the inlet, as illustrated in Fig. 1. Further details of the acoustic treatments and the measurement capabilities can be found in articles of past studies at Langley [18–20]. This curved duct is shaped to mimic an aft-fan duct in a turbo-fan engine. In this study, only results measured in the curved-duct configuration are discussed. Acoustic liners were installed in the 1.016 m (40 in) long test section to evaluate different liner configurations. The data analyzed in this study were obtained with a typical acoustic liner (perforated plate over honeycomb core) installed on the top and bottom sides of the curved test section in Fig. 1. Through this duct, unheated forced flow can be used to evaluate the effects of the free-stream Mach number on liner performance. The facility is capable of operating with mean-flow speeds up to Mach 0.5. Two identical arrays consisting of 47 flush-mounted piezo-ceramic microphones (1/4 in) are located upstream and downstream of the test section (see Fig. 1). Their microphone distribution was designed such that all the cut-on modes

Fig. 1. Schematic of duct geometry and microphone distributions (units are in inches). Speakers are installed on the left hand side, and flow and dominant acoustic signals propagate from left to right. In the laboratory, the duct is actually located such that the x- and y-axes are aligned in the vertical and horizontal directions, respectively.

Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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within the expected operational bandwidth (400–2500 Hz) can be resolved [18]. This pair of arrays can be used to calculate the acoustic attenuation of liners in the test section. In this frequency range, the number of the microphones, N mic ¼ 47, is always greater than the numbers of cut-on modes (i.e. over-determined problems); however, to examine the applicability of the beam-forming algorithms for under-determined problems, a subset of 10 microphones is also specified for selected cases (see the discussion in Section 4.1). An incident acoustic field was generated by 16 loud-speakers (not shown) located upstream of the test section and the microphone arrays. A specified mode can be excited at a single frequency potentially up to 2800 Hz using a computercontrolled system using adaptive feedforward that adjusts the phase and amplitude of each individual speaker's signal until the desired mode is maximized at the upstream microphone array [18]. In Table 1, the modes and their cut-off frequencies at the Mach numbers tested are listed. This capability was used to measure the effect of incident duct modes on linerattenuation performance. Measurements were performed with two types of acoustic excitation. The first set of measurements was made with mode-controlled tone excitation. For each run, one mode was driven at a single frequency, and the forcing frequency is increased from 400 Hz to 2800 Hz in 200 Hz increments. The numbers of cut-on modes at these frequencies are listed in Table 2. In this study, only the (0,0) mode excitation, which was stably controlled about up to 2000 Hz, is primarily analyzed. These data sets are referred to as ‘tone measurements’ in the rest of the paper. The second set of measurements was made with a broadband excitation of white noise from 300 to 3000 Hz, whereas the results up to 2000 Hz are discussed in this paper. This excitation was performed without using the mode-controlling system, and the number of excited modes was not specified. All the 16 loud-speakers were driven by a single-signal generator; hence, the sound was perfectly correlated upstream, and this will be observed in Section 4.2. These data are referred to as ‘broadband measurements’ in the rest of the paper. 3. Algorithms for duct-mode identification One of the major differences in beam-forming applications to duct acoustics is that the reference solutions are composed of discrete modes as opposed to Green's functions associated with a spatial domain for traditional source mapping/localization. Namely, a countable number of reference solutions can be identified with indices at a given frequency in a duct. Our interest is, for example, to extract duct-mode magnitudes associated with the driving signal for the purpose of evaluating the rate of attenuation for each mode. Therefore, we apply ‘beam-forming’ algorithms to a vector representing pressure signals in the frequency domain across the microphones instead of a cross-spectral matrix. Gerhold et al. [18] has adopted the mode-identification technique of this type for the mode-controlling system of this facility. The idea behind such pre-processing is as follows. Suppose that such a vector qorg containing complex pressure values across all the microphones at a given frequency is decomposed into qorg ≡α0 v0 þ ∑ αl′ vl′ þ w;

(1)

l′ ¼ 1

where the first term (with a subscript 0) denotes a signal associated with the driving signal (referred to as the primary signal), the second term is the superposition of coherent noise including components induced by interaction of the driving mode with flow motion, those independently generated by flow motion and the background noise, and the last term represents random noise uncorrelated between the microphones with mean zero. Here, we normalize the vectors, vl′ , on the

Table 1 Cut-off frequencies (by Hz) of modes that can be driven for tone excitation measurements. Mode: (m,n)

ð0; 0Þ

ð0; 1Þ

ð1; 1Þ

ð0; 2Þ

ð1; 2Þ

M ¼0.0 M ¼0.5

0 0

1128 976

1215 1052

2256 1953

2300 1992

Table 2 Number of cut-on modes as a function of frequency (by Hz). Upstream and downstream propagating modes are included. Frequency (Hz)

400

600

800

1000

1200

1400

1600

1800

2000

M ¼0.0 M ¼0.5

2 4

4 4

4 6

6 8

8 12

12 14

14 18

16 18

18 26

2200

2400

2600

2800

20 30

26 34

30 38

34 42

Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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right-hand side such that jvl′ j ¼ 1 for all l′ including 0 for convenience, and each coefficient αl′ then takes a positive real value with the unit equal to qorg . We simultaneously measure a reference time history synchronized with the driving signal and define its complex coefficient of the Fourier transform at the given frequency as qref , which is also normalized as jqref j≡1. Because our test cases include broadband signals, the phase of qref can vary for each time segment of the Fourier transform. By multiplying its complex conjugate with Eq. (1) and averaging it over many time segments, we derive q≡qnref qorg ¼ α0 qnref v0 þ ∑ αl′ qnref vl′ ;

(2)

l′ ¼ 1

where the random error vanishes as it is assumed to be uncorrelated with the driving signal. Because qref and v0 are perfectly correlated in principle, qnref v0 is a unit vector and jqnref v0 j ¼ 1. Therefore, if all the secondary noise components are completely uncorrelated with the driving signal, the magnitude of Eq. (2) yields α0 , extracting the amplitude of the primary signal; otherwise, it produces the superposition of the primary signal and the partial component correlated with the driving signal. In this study, we take the average of 1000 time segments for both tone and broadband measurements and 500 for synthetic-data analysis. Alternatively, a representative signal can be extracted from a cross-spectral matrix. Because it is Hermitian, a crossspectral matrix can be decomposed into Nmic

qorg q†org ≡ ∑ λl ul u†l ¼ UΛU† ;

(3)

l¼1

where conjugate transpose is expressed by †, a diagonal matrix, Λ, is composed of the eigenvalues, and a unitary matrix, U, is composed of their orthonormal eigenvectors (i.e. jul j ¼ 1 and u†l  um ¼ δlm ). It may be expected that the greatest eigenvalue pffiffiffiffiffi together with its eigenvector approximates the signal associated with the driving signal; namely, λ1 u1 ≈α0 v0 in Eq. (1). However, we should remember that the set of ul is generally different from the set of vl′ even if the number of vl′ is equal to or less than the number of microphones because the set of ul must be orthogonal to each other. In particular, when the pffiffiffiffiffi magnitude of α0 v0 is comparable to or less than that of the other modes, λ1 u1 substantially deviates from α0 v0 . We discuss this point later by using broadband measurement data. In this study, three representative approaches are considered: (i) conventional beam-forming (with steering vectors normalized based on least-squares minimization [21]), (ii) CLEAN [13], and (iii) the L1 generalized-inverse method [14] (or based on L2 norm [3–6]). Details of these original algorithms applied to cross-spectral matrices can be found in their previous articles. In the following, the procedure to apply these methods to aforementioned complex vectors extracted from a microphone array is summarized by focusing on the application to duct acoustics. Referring to the coordinates in Fig. 1, we expect that rectangular-duct modes represented by 7

pðm;nÞ ðx; y; zÞ ¼ cos ðkxðmÞ xÞ cos ðkyðnÞ yÞ expðikzðm;nÞ zÞ;

(4)

where

þ

kzðm;nÞ ≡

kM þ

 kzðm;nÞ ≡

kxðmÞ ≡

mπ ; Lx

(5a)

kyðnÞ ≡

nπ ; Ly

(5b)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 k ð1M2 ÞðkxðmÞ þ kyðnÞ Þ 1M 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 kM k ð1M 2 ÞðkxðmÞ þ kyðnÞ Þ 1M 2

;

(5c)

;

(5d)

are excited in the duct. Here, m and n denote the long-side and short-side mode orders, respectively, k the free-space 2 wavenumber (i.e. k ≡ω2 =c2 ), M ( o 1) the Mach number of the mean flow, and the superscripts + and  downstream- and upstream-propagating modes, respectively. In conventional beam-forming, complex amplitude aðm;nÞ for an individual mode is determined by minimizing J 2 ≡jqaðm;nÞ pðm;nÞ j2 ;

(6)

where q and pðm;nÞ respectively denote the measured pressure in the frequency domain, given by Eq. (2), and that of the specified mode, given by Eq. (4), in a vector form with the number of components equal to the number of microphones. By differentiating Eq. (6) with respect to each aðm;nÞ and setting it to be zero, the amplitude minimizing Eq. (6) is given by aðm;nÞ ¼

p†ðm;nÞ  q jpðm;nÞ j2

:

(7)

Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Thus, the amplitude, and the phase if necessary, of each mode can be determined from the vector consisting of Fourier-transformed pressure across the microphones, as opposed to a cross-spectral matrix. An output for a specified mode is independent of other modes in conventional beam-forming, and we must perform such calculation individually for all the modes of interest. Note that amplitude takes a non-zero value even for non-existing modes unless steering vectors of all the modes of interest are orthogonal to each other in conventional beam-forming, potentially causing significant contamination. The algorithm analogous to CLEAN [13] starts with the output of conventional beam-forming, Eq. (7), and iteratively stacks a fraction of a single-mode amplitude per iteration. To be concrete, we define the primary mode ðmðiÞ ; nðiÞ Þ as the one that gives the peak of jaðm;nÞ j at the ith iteration. We then record its output aðiÞ ¼ ðmðiÞ ;nðiÞ Þ

p†ðmðiÞ ;nðiÞ Þ  q ðiÞ jpðmðiÞ ;nðiÞ Þ j2

;

(8)

and subtract this contribution (one mode per iteration) from the Fourier-transformed pressure signal as q ðiþ1Þ ¼ q ðiÞ φaðiÞ p ðiÞ ðiÞ ; ðmðiÞ ;nðiÞ Þ ðm ;n Þ

(9)

where q ð1Þ ≡q and φ ð≤1Þ denotes the ‘loop gain’ introduced by Högbom [13]. Subsequently, the primary mode for the next iteration is again searched with Eq. (8), and its amplitude aðiþ1Þ is similarly recorded (thus, ðmðiÞ ; nðiÞ Þ can vary at each ðmðiþ1Þ ;nðiþ1Þ Þ iteration). This operation is repeated until jq ðiþ1Þ j becomes greater than jq ðiÞ j or the iteration counter reaches a designated number, which is set to be 300 in this study. The amplitude of each ðm; nÞ mode eventually yields the superposition of the complex amplitudes at the iterations only when this mode takes the maximum: φ

only for the same ðm;nÞ

∑ i

aðiÞ : ðmðiÞ ;nðiÞ Þ

(10)

In this study, the loop gain is set to be φ ¼ 0:5, which is relatively small in order to capture secondary modes with lower sound power. For CLEAN-SC, all the signals correlated with the primary mode are subtracted in Eq. (9), and the vector corresponding to pðmðiÞ ;nðiÞ Þ can be readily deduced from Eq. (26) in Sijtsma [11]. However, this approach is not beneficial when we process correlated pressure signals in a vector form because it removes all the secondary modes correlated with the primary one and detects only the amplitude of the primary mode. On the other hand, the idea behind the generalized-inverse method is to inversely estimate the complex modeamplitude distribution that recovers the Fourier-transformed signal across the microphones based on the Lp norm (p≤2). Namely, the Lp generalized-inverse algorithm seeks the mode distribution that minimizes the following cost function with a Lagrangian multiplier vector λ: J p ≡jajp þ

jwj2 þ λ†  ðqAawÞ; ϵ

(11)

for an under-determined problem, and J p ≡jqAaj2 þ ϵjajp ;

(12)

for an over-determined problem, again where a denotes the mode-amplitude distribution in a vector form, w measurement noise (the number of components being the number of microphones), q the aforementioned Fourier-transformed pressure signal given by Eq. (2), ϵ the regularization parameter and A the transfer function from a specified mode to a signal at a microphone. For duct acoustics, Aa should be replaced by ∑m ∑n aðm;nÞ pðm;nÞ , by following the aforementioned notation. Measurement uncertainty, w, is not explicitly included in the over-determined equation, as it is treated as a part of the cost function itself to be minimized. The first term above appears to be similar to Eq. (6), but the one in Eq. (12) includes multiple-mode amplitudes, and the algorithm solves them simultaneously. We should note in Eq. (12) that the Lp norm is applied to the amplitude distribution (i.e. the second term) as opposed to the constraint equation (i.e. the first term) even for an over-determined problem. Amplitudes of all the modes are then simultaneously computed using the generalized-inverse method. Because the cost function is decreased by collecting coherent sources rather than by distributing them in space for traditional source mapping, contamination from other modes can be suppressed and the resolution (i.e. dynamic range) is improved by decreasing Lp from L2 to L1 (refer to [14] for discussion of the benefit of an L1 algorithm). The actual calculation, which is summarized below, was performed with p¼ 1 [22]. Unlike the original generalized-inverse beam-forming algorithm [14], the number of modes to be processed is unchanged during iteration (i.e. the reduction factor, β in [14], is not introduced) because the number of duct modes is fairly limited compared with the number of grids for typical source mapping. Thus, the procedure of the Lp generalizedinverse algorithm applied to this study, actually p¼1, can be listed as follows: (1) Fourier-transform time-histories of microphone signals, multiply them by the Fourier-transformed reference signal, and average them (denoted by q≡qnref qorg Þ. (2) Define the numbers of long-side and short-side modes ðm; nÞ to be processed, calculate a transfer matrix, A, consisting of their mode shapes, and store it. Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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(3) Calculate the initial mode distribution, a, using a≈ðA† A þ ϵIÞ1 A† q if Nmic 4 Nmode . (If N mic o Nmode , calculate a using a≈A† ðAA† þ ϵIÞ1 q.) (4) Improve the dynamic range by solving a with aðiþ1Þ ≈ðA† A þ ϵðW ðiÞ Þ1 Þ1 A† q if N mic 4N mode (or with aðiþ1Þ ≈W ðiÞ A† ðAW ðiÞ A† þ ϵIÞ1 q, otherwise), where W ðiÞ is a diagonal matrix with components of w≡jaðiÞ j2p . (5) Iterate (4) until the Lp norm, jajp , starts increasing or the iteration counter i reaches a designated number, set to be 10 in this study. (6) Generate a mode distribution at each frequency. For the L2 generalized-inverse (or pseudo-inverse) method, the procedure is stopped up to (3) above. The iterative procedure in (4) is referred to as an Iteratively Re-weighted Least-Squares (IRLS) technique [22]. The artificial diagonal parameter, ϵ, is set to be 0.01, which is found to be effective over a wide frequency range (some discussion regarding the regularization parameter is available in [23]), and the cut-on modes up to 0≤m≤8 and 0≤n≤3 propagating in both axial directions are processed throughout this study to cover all the cut-on modes at the frequencies of our interest. 4. Results and discussion 4.1. Synthetic-data analysis To compare the accuracy of conventional beam-forming, CLEAN as well as L1 and L2 generalized-inverse beam-forming, the averaged error and the standard deviation of the detected amplitude, which are relevant to diagnostics, are first evaluated using synthetic data in the test-facility configuration. To prescribe a signal to be detected, a driving signal consisting of five different downstream-propagating modes, (m, n) ¼ (0, 0), (1, 0), (2, 0), (0, 1), and (1,1), given by Eq. (4), is imposed by setting the magnitudes successively decreasing by 2 dB with random phases. In addition, 10 percent random noise (  10 dB relative to the (m, n) ¼ (0, 0) mode, with uniform magnitude across the microphones) is added with random phases. Thus, an attempt is made to realistically mimic the CDTR test environment. Statistics over 500 samples are taken, and amplitudes are calculated for all the cut-on modes that are propagating both upstream and downstream. In the following tests, we assume M¼0 and the driving frequency of 1600 Hz, in which the number of cut-on modes is 14 as listed in Table 2, and consider two microphone arrays: One consists of the original Nmic ¼ 47 microphones and the other a subset of N mic ¼ 10 microphones, whose coordinates are illustrated in Fig. 2. Consequently, the problem yields overdetermined for the former case and under-determined for the latter in the generalized-inverse method. In this study, an under-determined condition is hypothetically created, yet it is important in many practical applications in which the number of microphones/sensors is limited relative to the number of cut-on modes, particularly at high frequencies. Fig. 3 compares the errors/deviations in detected amplitude of each driven mode across the four algorithms with N mic ¼ 47 microphones. Although the averaged detected levels in conventional beam-forming are close to the expected values, the standard deviations in detected amplitude vary by one or two decibels depending on the forced modes. On the other hand, CLEAN and both generalized-inverse approaches improve the accuracy: Their standard deviations decrease by an order of magnitude, and the noise levels other than the driven modes (regarded as noise floor, whose values are noted in the caption) also become smaller. Greater dynamic range is useful for mode identification, for example in spectrograms. For the over-determined condition, the detected amplitude is nearly unchanged from the L2 to L1 algorithm. In the under-determined condition using Nmic ¼ 10 microphones, as displayed in Fig. 4, the standard deviations of all the algorithms increase significantly compared with those using N mic ¼ 47. Conventional beam-forming generally shows the largest

y

z x Fig. 2. Reduced microphone distribution to test under-determined problems (units are inches). Symbols: ○, original microphones (N mic ¼ 47); n, reduced microphone system (N mic ¼ 10).

Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

5

5

0

0

-5

-5

-10

-10

-15

-15

-20

-20

7

-25

-25 (0,0)

(1,0)

(2,0)

(0,1)

(1,1)

5

5

0

0

-5

-5

-10

-10

-15

-15

-20

-20

(0,0)

(1,0)

(2,0)

(0,1)

(1,1)

(0,0)

(1,0)

(2,0)

(0,1)

(1,1)

-25

-25 (0,0)

(1,0)

(2,0)

(0,1)

(1,1)

Fig. 3. Errors of the mode amplitude detected using 47 microphones for M ¼ 0 at 1600 Hz. A box denotes the standard deviation of errors in a decibel scale with a dashed horizontal line at the center indicating the averaged error. A dotted vertical line ranges from the minimum to the maximum values among 500 samples for each mode. (a) Conventional beam-forming (  21 dB). (b) CLEAN (  45 dB). (c) L2 generalized inverse (  40 dB). (d) L1 generalized inverse (  50 dB). Numbers inside the parentheses are the averaged residual noise levels (i.e. noise floor), and they are drawn by a solid line if within the plotted range.

standard deviations, but the averaged errors are the smallest. The other three algorithms tend to under-estimate the amplitude, yet with much lower noise floor than that of conventional beam-forming: CLEAN and L2 generalized-inverse beam-forming are generally less accurate than L1 generalized-inverse beam-forming (to be precise, when the forced amplitude approaches the 10 percent noise level, the accuracy of the L1 algorithm rapidly deteriorates, and CLEAN is sometimes unable to identify the mode). Because mode distributions that recover the received signal at the microphones are no longer unique, the L2 algorithm tends to select a distribution that spreads the amplitudes across the prescribed modes, leading to smaller jaj2 . This possibly explains the reason for significant under-estimation of the amplitude in Fig. 4(c). As a result, L1 generalized-inverse beam-forming demonstrates the benefit over the L2 algorithm for the under-determined condition. In the next test, synthetic signals including the 10 percent noise are similarly generated, starting with one mode and progressing up to six cut-on modes driven in the order noted before (the last mode is (m, n) ¼ (3,0) in the order of the axial wavenumber) with random phases. To evaluate the accuracy of acoustic-field reconstruction, Fourier-transformed pressures at the positions corresponding to all 47 microphones in the CDTR arrays are recovered from the detected mode amplitudes and phases. From such a test, accuracy of an acoustic field propagated elsewhere from the duct may be assessed, for example from an engine to a far field. Errors of the reconstructed pressure signal from the original pressure in a decibel scale are then summed over all the microphones and averaged over 500 samples, denoted by an over-line, as 1 Nmic ð ¼ 47Þ sig ΔSPL≡ ∑ ðSPLrec l SPLl Þ; Nmic l ¼ 1

(13)

s2 ðΔSPLÞ≡ðΔSPLΔSPLÞ2 ;

(14)

and its standard deviation is defined as

Figs. 5 and 6 plot such errors/standard deviations using N mic ¼ 47 and 10 microphones for detection, respectively. With both microphone counts, we observe that conventional beam-forming suffers from a systematic over-estimate even with Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Fig. 4. Errors of the mode amplitude detected using 10 microphones for M ¼0 at 1600 Hz. (a) Conventional beam-forming (  14 dB). (b) CLEAN (  26 dB). (c) L2 generalized inverse (  24 dB). (d) L1 generalized inverse (  31 dB). Notation is the same as Fig. 3.

one mode driven. While the conventional method can detect the correct amplitude of a single mode in a statistical sense, detected amplitudes of unforced modes are not negligible. Such contamination is caused by projection of a signal from the driven mode onto components of unforced modes due to non-orthogonality of the steering vectors. When the amplitudes from all modes are summed, errors are accumulated by these unforced modes, causing over-estimate of the overall amplitude. The other three approaches can recover the sound pressure levels at the microphone positions better than conventional beamforming. These algorithms can detect insignificant magnitudes for unforced modes, leading to a higher dynamic range. In particular, L1 generalized-inverse beam-forming can retain the accuracy better than or equal to the other algorithms for both over- and underdetermined conditions. CLEAN sometimes misses detecting driven modes; however, the accuracy of detection is equally well with the L2 algorithm, and the contribution from unforced modes is least, as discussed later. Although both CLEAN and generalized-inverse approaches demonstrate partly better accuracy over conventional beamforming in identifying mode-amplitude distribution and in recovering pressure signals at the microphone positions, the situation above is still idealized. To simulate broader practical situations, the aforementioned random-phase noise is increasingly added to the synthetic pressure signals composed of the five prescribed cut-on modes, and the sensitivity/ robustness to extraneous noise is evaluated for conventional beam-forming, CLEAN and L1 generalized-inverse beamforming below. Fig. 7(a) plots 300 samples of errors in detected amplitude of the primary mode, i.e. ðm; nÞ ¼ ð0; 0Þ, for M ¼0 at 1600 Hz as a function of the normalized noise level given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ∑Nmic jpnoi j2 Nmic l ¼ 1 l (15) ðNormalized noise levelÞ≡ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 sig 2 mic ∑N 9p 9 N mic l ¼ 1 l and psig denote pressure fluctuations of the noise and the original signal, respectively, at the lth microphone where pnoi l l position. Thus, the noise level ranges approximately from 14 dB to +14 dB relative to the original signal as shown in Fig. 7 Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Fig. 5. Comparison of errors in reconstructed pressure using N mic ¼ 47 at all the microphone positions versus the number of modes driven for M ¼0 at 1600 Hz. (a) Conventional beam-forming. (b) CLEAN. (c) L2 generalized inverse. (d) L1 generalized inverse.

such that it mostly covers the range of the CDTR test environment. At lower noise levels, CLEAN and generalized-inverse beam-forming detect the primary-mode amplitude equally well as or somewhat better than conventional beam-forming, but their deviations spread more rapidly with increasing noise level. In particular, CLEAN occasionally misses detecting the primary mode (detected amplitude less than 30 dB from the original signal is plotted at the  30 dB line), while L1 generalized-inverse beam-forming retains the distribution close to the 0 dB level. We should note that the L1 and L2 algorithms show no significant difference for this condition (not shown). Fig. 7(b) shows that CLEAN tends to recover the pressure signals at the microphone positions accurately against higher noise levels. Although CLEAN sometimes misses one or a few driven modes, the rest of the modes recover the original pressure signal very well. In contrast, conventional beam-forming substantially over-estimates the mode amplitude due to contamination from unforced modes, as mentioned before. Even at higher noise levels, L1 generalized-inverse beamforming can detect the driven modes most accurately but also includes minor amplitude distribution across the unforced modes, which attempts to recover the noise components; as a result, its accuracy in the recovered pressure signal is somewhat worse than that of CLEAN yet still better than that of conventional beam-forming even when the noise level exceeds the magnitude of the signal. Even by detecting with only 10 microphones, the trends of the three methods remain the same in Fig. 8. CLEAN fails to detect the primary mode more often compared to Fig. 7, and its errors scatter somewhat greater than L1 generalized-inverse beam-forming at higher noise levels. CLEAN still best recovers the original pressure signal at the microphone positions over the entire noise levels, while conventional beam-forming always over-estimates it. These results conclude that generalized-inverse beam-forming and CLEAN are less sensitive to extraneous noise than conventional beam-forming as long as the noise level is less than or comparable to the magnitude of the original signal. For CLEAN, smaller loop gain improves the accuracy a little but substantially increases computational cost. In this benchmark problem, CLEAN requires an order of 100 iterations and its computational cost is about three times more expensive than that of L1 generalized-inverse beam-forming with 10 iterations. Moreover, CLEAN sometimes misses even the primary mode. These observations imply that L2 generalized-inverse beam-forming is practically most advantageous for over-determined

Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Fig. 6. Comparison of errors in reconstructed pressure using N mic ¼ 10 at all the microphone positions versus the number of modes driven for M ¼ 0 at 1600 Hz. (a) Conventional beam-forming. (b) CLEAN. (c) L2 generalized inverse. (d) L1 generalized inverse.

problems because of its low computational cost, and L1 for under-determined problems among the approaches that we tested in terms of accuracy, robustness, and computational cost.

4.2. CDTR measurement analysis Now, the three methods, conventional beam-forming, CLEAN and L1 generalized-inverse beam-forming, are applied to the test data acquired at the NASA CDTR facility. The results of L2 generalized-inverse beam-forming are omitted for overdetermined problems because they are essentially unchanged from those of L1 generalized-inverse beam-forming. As mentioned in Section 3, Fourier-transformed pressure signals are pre-multiplied by the reference signal to increase the signal-to-noise ratio in pre-processing. Since the dominant flow noise is considered to be incoherent with the driving signal, its contamination is substantially suppressed after averaging over many time segments. This is necessary since the flow noise contamination inside the duct can be large at higher Mach numbers. For the following discussion, we define the acoustic intensity (i.e. power) in the axial flow by following the discussion by Morfey [24] as 2

2

I ðm;nÞ ≡jaðm;nÞ j2 ½MðkxðmÞ I sc þ kyðnÞ I cs jkzðm;nÞ j2 I cc þ jkkzðm;nÞ Mj2 I cc Þ þ2k Reðkzðm;nÞ ÞI cc =ð4ρcLx Ly jkkzðm;nÞ Mj2 Þ;

(16)

where OðM 2 Þ order terms are ignored, and Z I sc ≡

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if m ¼ 0 if m≠0; n ¼ 0

;

(17a)

otherwise

Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Z I cs ≡

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In Fig. 9, the mode-power distributions in the upstream sector are compared across the three methods for tone measurement at several different frequencies. Here, flow was deactivated (i.e. M¼0.0), and plane waves, i.e. the ðm; nÞ ¼ ð0; 0Þ mode, were forced. Although an attempt is made to control the amplitude of the driving mode using a feedforward system, the accuracy above 2000 Hz, indicated by a shaded region, is uncertain because of the limited number of the loud-speakers and the system time response. Thus, the results below 2000 Hz, in which the conditions are all overdetermined (refer to Table 2 for the numbers of cut-on modes), are mainly discussed below. In these figures, both upstream and downstream propagating modes are plotted, but only for representative cut-on modes. The detected magnitude of the forced mode is similar in all three methods; however, the magnitudes of the unforced modes in conventional beam-forming tend to be much higher than those in CLEAN and L1 generalized-inverse beamforming. While conventional beam-forming extracts the ð0; 0Þ mode about 10 dB or less above other unforced modes, the other two algorithms clearly enhance a dynamic range up to about 1600 Hz. This suggests that the unforced-mode Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Fig. 9. Comparison of the mode-power distributions in the upstream sector produced from the test data (M ¼ 0.0). The ð0; 0Þ mode was forced. Large symbols denote downstream-propagating modes and small symbols upstream-propagating modes. (a) Conventional beam-forming. (b) CLEAN. (c) L1 generalized-inverse beam-forming.

magnitudes have reached the noise floor of the conventional beam-forming technique. On the other hand, the other two methods exhibit almost an identical mode distribution including first few unforced modes. The increased dynamic range by these two algorithms allows more accurate measurement of mode-power distribution even beyond 10 dB below the dominant mode. Likewise, Fig. 10 plots the mode distributions in the downstream sector. The detected intensity of the driving mode ð0; 0Þ is about the same across the three methods. CLEAN and generalized-inverse beam-forming again produce similar mode distributions and appear to retain much higher dynamic range than that of conventional beam-forming. From 1200 to 1600 Hz, the intensity of an unforced mode ð0; 1Þ becomes comparable to the driving mode regardless of the algorithms. This suggests that this mode physically propagates within the test section, and its magnitude is not an artifact of detection/ deconvolution inaccuracy. Although the long-side mode ð1; 0Þ becomes cut-on first with increasing frequency, the excitation of the short-side mode ð0; 1Þ turns out to be more relevant probably due to the curve of the duct (see Fig. 1). The increased dynamic range of CLEAN and generalized-inverse beam-forming resolves other unforced modes more than 10 dB below the ð0; 1Þ mode, indicating its benefit. Validity of the mode-identification techniques in both upstream and downstream sectors assures that liner performance can be evaluated using this facility by referring to the ratio of the mode power between the two sectors. Even when the mean flow of M ¼0.5 is activated in the test section, the trend is about the same. Figs. 11 and 12 similarly compare the detected mode-power distributions in the upstream and downstream sectors, respectively. The dynamic range appears to even increase, and CLEAN and generalized-inverse beam-forming secure about a 20 dB dynamic range, at least, up to 1400 Hz both upstream and downstream. We should note here that the number of cut-on modes increases due to the mean flow, as shown in Table 2, and this can challenge the forcing system at higher frequencies. Interestingly, the downstream distribution in Fig. 12 again indicates generation of the ð0; 1Þ mode at 1200–2000 Hz regardless of the algorithms, supporting the aforementioned hypothesis. These results demonstrate that the two algorithms are effective even when high-speed mean flow exists. We should note an important factor which may be deteriorating the accuracy of mode detection. In the algorithms examined here, we prescribe only cut-on modes to be detected. However, some cut-off modes have a relatively slow decay Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

13

Fig. 10. Comparison of the mode-power distributions in the downstream sector (M¼ 0.0). Conditions and notations are the same as Fig. 9. (a) Conventional. (b) CLEAN. (c) L1 generalized-inverse.

rate, and the number of such modes tends to increase with increasing frequency. In the current configuration, the imaginary 7 part of the axial wavenumber up to about jIm½kz j≤10 (m  1) seems to somewhat influence the mode-power distribution. It is non-trivial to process these ‘slowly decaying’ modes because the amplitude strongly depends on their origin. We consider this issue in the scope of future study. To observe the benefit in minimization of L1 norm over L2 norm for an under-determined problem, we process the same data set using N mic ¼ 10 microphones illustrated in Fig. 2. Fig. 13 plots the mode-power distributions comparing the L2 and L1 generalized-inverse beam-forming algorithms in the upstream sector at M¼0.5. We should pay attention to the range from 1200 to 2000 Hz, in which the condition is under-determined but the feedforward speaker-control system is still capable of resolving the forcing mode. Compared with the distribution based on the L2 norm, the L1 algorithm clearly enhances the dynamic range at 1200 Hz and 1400 Hz, which is nearly equivalent to the full microphone system shown in Fig. 11(c). Even at 1600 Hz, the noise floor seems to be suppressed as the ð1; 1Þ mode can be recognized from Fig. 11 as a secondary mode actually propagating. In the downstream sector, plotted in Fig. 14, the ð0; 1Þ mode is induced above 1200 Hz in addition to the ð0; 0Þ mode, but the amplitudes of the other modes are detected much less in L1 generalized-inverse beam-forming. Thus, improvement in the dynamic range based on the L1 norm is similarly observed downstream. From this extra test, the benefit of the L1 algorithm against the L2 algorithm can be confirmed for under-determined conditions, although the effective frequency range is somewhat limited for an array with a large number of microphones. It also implies that the full array system can potentially resolve the mode-power distribution at much higher frequencies if the forcing system were to be able to respond properly. Now we analyze the data of broadband measurement in which a single white-noise signal with a frequency range of 300–3000 Hz was branched and input to all 16 speakers. Before displaying the mode-power distributions, we observe the effect of pre-processing using the reference signal, discussed in Section 3. As an example, broadband signals with M¼0.5 mean flow at 1800 Hz are processed, and the extracted vector, q given by Eq. (2), is compared with the eigenvector of the cross-spectral matrix, given by Eq. (3). Fig. 15(a) depicts the eigenvalue distribution in the upstream sector together with jqj2 denoted by a horizontal line. The cross-spectral matrix has a single large eigenvalue, indicating strong coherence across the microphones. The vector q Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Fig. 11. Comparison of the mode-power distributions in the upstream sector produced from the test data (M ¼ 0.5). The ð0; 0Þ mode was forced. Large symbols denote downstream-propagating modes, and small symbols upstream-propagating modes. (a) Conventional beam-forming. (b) CLEAN. (c) L1 generalized-inverse beam-forming.

Fig. 12. Comparison of the mode-power distributions in the downstream sector (M¼0.5). Conditions and notations are the same as Fig. 11. (a) Conventional. (b) CLEAN. (c) L1 generalized-inverse.

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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extracted using the reference signal recovers about 76 percent of the largest eigenvalue by the squared value. In fact, excellent agreement of the normalized vectors in Fig. 15(b) proves that the driving signal dictates the coherent acoustic field in the upstream sector. For the M¼0 case, coherency is similarly strong in both upstream and downstream sectors (not shown). In contrast, the eigenvalue distribution downstream at M ¼0.5 spreads widely in Fig. 16(a), and the extracted signal poorly recovers the sound power. Fig. 16(b) clearly shows that the first three eigenvectors totally miss the phase relation of the extracted signal. In general, coherency is lower at higher frequencies. Thus, we speculate that noise generated purely by flow motion dominates the acoustic field in the downstream sector, and the approach introduced here cannot extract signals associated with the driving signal effectively. In the following analysis of the broadband measurement, we only discuss the results from the upstream sector with Nmic ¼ 47. Fig. 17 compares the mode-power distributions at M¼0.0 across the three methods. CLEAN and L1 generalized-inverse beam-forming produce very similar spectral shapes for first few dominant modes, while conventional beam-forming agrees with them only for the primary mode at each frequency (i.e. ðm; nÞ ¼ ð0; 0Þ at low frequencies and ð2; 0Þ at high frequencies). It is fair to deduce that the dynamic ranges of those two methods are better than that of conventional beam-forming by as much as 10 dB up to about 1500 Hz. In particular, the magnitudes of upstream-propagating modes substantially decrease in those two methods, as expected from the open-end downstream condition of the test facility. Even if we activate the mean flow of M¼ 0.5 in the broadband measurement, Fig. 18 demonstrates that CLEAN and generalized-inverse beam-forming can retain better dynamic range than conventional beam-forming. In fact, the power ratio between the downstream- and upstream-propagating modes becomes somewhat greater than that in the no-flow case, probably owing to mean-flow convection. CLEAN intermittently drops the mode power of weaker modes, and such a trend is consistent with the analysis in Section 4.1. Similar to the no-flow case in Fig. 17, the primary mode changes from

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Fig. 13. Comparison of the mode-power distributions in the upstream sector with only 10 microphones (M ¼0.5). Conditions and notations are the same as Fig. 11. Problem becomes under-determined above 1170 Hz, denoted by a dotted vertical line. (a) L2 generalized-inverse. (b) L1 generalized-inverse.

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Fig. 14. Comparison of the mode-power distributions in the downstream sector with only 10 microphones (M ¼ 0.5). Conditions and notations are the same as Fig. 13. (a) L2 generalized-inverse. (b) L1 generalized-inverse.

Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

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Fig. 15. Comparison between the eigenmode and the extracted signal in the upstream sector (1800 Hz and M ¼0.5). (a) Eigenvalue distribution. A horizontal line denotes jqj2 . (b) Normalized vectors in the complex plane: ○, first eigenvector; n, extracted signal.

Fig. 16. Comparison between the eigenmode and the extracted signal in the downstream sector (1800 Hz and M ¼ 0.5). Conditions and notations are the same as Fig. 15. (a) Eigenvalue distribution. (b) Normalized vectors in the complex plane: ○, first eigenvector; ⋄, second eigenvector; □, third eigenvector; n, extracted signal.

ðm; nÞ ¼ ð0; 0Þ to ð2; 0Þ with increasing frequency. These results indicate that all the approaches can actually track transition of mode changes upstream even in broadband-noise environment regardless of the presence of the mean flow. In the past studies [7,8], conventional beam-forming was used to identify modes of broadband noise from an aeroengine, particularly for spinning modes using a ring array. Presumably, other approaches were avoided because the number of modes to be detected tends to be greater than the number of microphones. Although the current configuration is very different from these studies, this study indicates the potential of CLEAN and L1 generalized-inverse beam-forming for broadband-noise diagnosis. Both showed better dynamic ranges compared with conventional beam-forming for broadband noise even in under-determined conditions. Compared with actual aero-engines, free-stream turbulence intensity is lower in CDTR [19] because of a honeycomb and several screens upstream. Hence, the background noise potentially makes mode identification more challenging in real engine tests. In addition, the blade counts dictate the number of azimuthal modes to be resolved, and they are far more than the number of rectangular modes in this facility. Therefore, gaining higher dynamic range is more important in practical aero-engine tests.

5. Conclusions Three approaches, conventional beam-forming, CLEAN and generalized-inverse beam-forming, which are typically used for source mapping/localization, have been examined to identify rectangular-duct acoustic modes from phasedarray pressure signals. These algorithms have been applied to a vector representing a signal correlated with the driving mode, as opposed to a cross-spectral matrix. Namely, the algorithms have been tested when the acoustic field is Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Fig. 17. Comparison of the mode distributions in the upstream sector for the broadband measurement. Configured with acoustic liner at M¼ 0.0. Thicker lines denote downstream-propagating modes, and thinner lines upstream-propagating modes. (a) Conventional. (b) CLEAN. (c) L1 generalized-inverse.

composed of multiple coherent modes associated with a recordable driving signal, representing an engine test for an aircraft, for example. In particular, the generalized-inverse method has been formulated for both over-determined (N mic 4 Nmode ) and under-determined (Nmic oN mode ) problems, and the algorithms based on L2 and L1 norms have been evaluated. Fundamental features of the algorithms have been first tested using synthetic data mimicking the CDTR geometry at NASA Langley Research Center. Even when multiple cut-on modes are activated, higher dynamic ranges of the detected modes can be secured by CLEAN and generalized-inverse beam-forming than by conventional beam-forming. While the problem is over-determined using all 47 microphones, L2 and L1 generalized-inverse algorithms can produce mode distribution equally well compared with CLEAN. When the condition becomes under-determined, hypothetically using fewer microphones, the accuracy of L2 generalized-inverse beam-forming starts deteriorating. When we attempt to recover the acoustic field from the detected mode distribution, it is clear that conventional beamforming suffers from over-estimation of amplitude for non-existing modes. The other two approaches, CLEAN and generalized-inverse beam-forming, essentially find the mode distribution that best recovers the signal at the microphone positions, resulting in less contamination from non-existing modes. Synthetic-data analysis with extraneous noise has also confirmed that these two methods are more robust than conventional beam-forming, so long as the noise level is equal to or less than the magnitude of the signal of our interest. Subsequently, test data acquired at CDTR have been processed by conventional beam-forming, CLEAN and L1 generalizedinverse beam-forming, and the results have been compared. Propagating cut-on modes have been identified both upstream and downstream of the test section with and without the mean flow of M¼0.5. To drive the signals, two types of excitation have been performed: (i) plane waves, i.e. ð0; 0Þ mode, have been driven at a target frequency by a mode-generating feedforward system; and (ii) coherent white-noise has been excited. In both cases, better dynamic range is secured by CLEAN and L1 generalized-inverse beam-forming over a range of frequencies relative to conventional beam-forming, although the magnitude of the primary mode (sometimes first few of them) is detected equally well by all three approaches. Capability of detecting secondary modes (i.e. non-driving ones) can be inferred from agreement of some lower modes across the three methods. Thus, this study has demonstrated that CLEAN and the generalized-inverse method can resolve acoustic duct modes equally well each other and better than conventional beam-forming in terms of amplitude detection of the primary mode Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

T. Suzuki, B.J. Day / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Fig. 18. Comparison of the mode distributions in the upstream sector for the broadband measurement. Configured with acoustic liner at M¼ 0.5. Notations are the same as Fig. 17. (a) Conventional. (b) CLEAN. (c) L1 generalized-inverse.

and the dynamic range. The first two methods are computationally more expensive than conventional beam-forming, but the number of modes for duct acoustics is generally limited compared with the number of grid points for typical noisesource mapping/localization. Hence, the actual computational cost for the mode-identification part is much lower than that for pre-processing of time-history data using the Fourier transforms. Since the accuracy of CLEAN and generalized-inverse beam-forming is nearly equivalent, the L2 generalized-inverse algorithm, which requires no iteration, is considered to be most beneficial for over-determined problems, while the L1 algorithm, which tends to be still less expensive than CLEAN, is most practical for under-determined problems.

Acknowledgments We gratefully acknowledge the support by NASA Langley Research Center for this project, particularly Dr. Carl H. Gerhold. We also would like to thank Dr. Cyrille C. Breard, who led this project previously, and Dr. John Rose in the Boeing Company for many useful comments. References [1] P. Joppa, An acoustic mode measurement technique, AIAA Paper, AIAA-84-2337, 1984. [2] J. Premo, P. Joppa, Fan noise source diagnostic test-wall measured circumferential array mode results, AIAA Paper, AIAA-2002-2429, 2002. [3] Y. Kim, P.A. Nelson, Estimation of acoustic source strength within a cylindrical duct by inverse methods, Journal of Sound and Vibration 275 (2004) 391–413. [4] T. Schultz, L.N. Cattafesta, M. Sheplak, Modal decomposition method for acoustic impedance testing in square ducts, Journal of Acoustical Society of America 120 (6) (2006) 3750–3758. [5] U. Tapken, L. Enghardt, Optimisation of sensor arrays for radial mode analysis in flow ducts, AIAA Paper, AIAA-2006-2638, 2006. [6] T. Bravo, C. Maury, Enhancing the reconstruction of in-duct sound sources using a spectral decomposition method, Journal of Acoustical Society of America 127 (6) (2010) 3538–3547. [7] C.R. Lowis, P.F. Joseph, A.J. Kempton, Estimation of the far-field directivity of broadband aeroengine fan noise using an in-duct axial microphone array, Journal of Sound and Vibration 329 (2010) 3940–3957. [8] P. Sijtsma, Using phased array beamforming to identify broadband noise sources in a turbofan engine, International Journal of Aeroacoustics 10 (1) (2011) 39–52. [9] P. Sijtsma, Feasibility of in-duct beamforming, AIAA Paper, AIAA-2007-3696, 2007.

Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i

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Please cite this article as: T. Suzuki, & B.J. Day, Comparative study on mode-identification algorithms using a phasedarray system in a rectangular duct, Journal of Sound and Vibration (2013), http://dx.doi.org/10.1016/j.jsv.2013.06.027i