By starting from a large initial array stencil, the microphone ... microphone removal and an accelerated procedure that removes several microphones per it-.
Acoustic beamforming array design using an iterative microphone removal method Elias J. G. Arcondoulis∗ and Yu Liu† Southern University of Science and Technology (SUSTech), Shenzhen, Guangdong Province, China, 518055 A well-designed acoustic beamforming array aims to simultaneously minimise the Maximum Sidelobe Level (MSL) and Main Lobe Width (MLW) within a desired acoustic frequency range, yet these properties tend to be mutually exclusive. This paper describes an iterative microphone removal method. By starting from a large initial array stencil, the microphone within the array that results in the smallest product of frequency-averaged MLW and MSL of the point spread function is identified and removed. This process is then repeated until a predetermined desired number of microphones is reached. Using the cross-spectral beamforming algorithm, several 841 and 961-channel equi-and-non-equispaced grid arrays are reduced to unique 48-channel arrays. The reduction method is implemented in two ways, i.e., one-by-one microphone removal and an accelerated procedure that removes several microphones per iteration, where the removed number of microphones decreases in an exponential manner with increasing iterations. The reduced arrays are compared against several logarithmic spiral and randomised 48-channel pattern arrays using a non-dimensional metric, evaluated as the product of the MSL normalised by main lobe magnitude, and the ratio of the MLW area to the total scanning grid area. The frequency range of investigation is 2000 Hz to 8000 Hz to replicate typical conditions in small scale anechoic facilities. The array reduction methods reveal improved array designs according to the aforementioned metric.
I. Nomenclature α β, γ δ δmin δav δmax λ ̂ υ ϕ Φ Φav Φmin a,b c C E f fmin fmax fs l m ⃗ m
= = = = = = = = = = = = = = = = = = = = = = =
number of microphones removed one at a time exponential decay profile parameters distance between microphones minimum distance between any two microphones within an array mean distance between all microphones within an array maximum distance between any two microphones within an array acoustic wavelength steering vector between microphone and scanning grid locations spiral sweep angle non-dimensional beamformer quality metric frequency-averaged Φ minimum single-frequency value of Φ expansion / contraction coefficients of the logarithmic spiral equation speed of sound in air cross-spectral matrix exponential decay profile acoustic beamforming frequency minimum acoustic frequency of interest maximum acoustic frequency of interest number of frequencies to be investigated between fmin and fmax number of sweeps of ϕ microphone number linearly spaced vector of microphones
∗ Postdoctoral † Associate
Research Associate, Department of Mechanics and Aerospace Engineering, AIAA Member Professor, Department of Mechanics and Aerospace Engineering, AIAA Senior Member
M Mi Mf n N N3dB p r rm rs w Y x,y,z
A
= number of microphones in an array = number of microphones in initial array stencil = number of microphones in the desired array = grid-array design parameter = number of scanning grid points = number of scanning grid points covered by the main lobe = complex acoustic pressure vectors = microphone array coordinates = distances from grid points to microphones = distances from the source point to microphones = shading vector = cross-spectral beamformer output = local coordinate directions
II. Introduction
coustic beamforming is a widely used and popular experimental technique that utilises a phased array of microphones to locate and quantify acoustic sources, which has been used extensively in the investigations of airfoil trailing edge noise [1–6], aircraft landing gear and fuselage noise [7–10] and wind turbine noise [11, 12]. Delay-and-sum and conventional beamforming [13] is a fast and simple method yet the source image maps can be contaminated with spatial aliasing images called sidelobes. The magnitude of the greatest sidelobe is referred to as the maximum sidelobe level (MSL). Removal of the autospectra components, namely cross-spectral beamforming (CSB) (or also known as conventional beamforming) can improve the noise source resolution in certain applications [14]. A well-designed array can help reduce both the MSL and the main lobe width (MLW). Logarithmic spiral array patterns have been shown to produce efficient array patterns for a given number of microphones and allowable array area [15, 16]. Using simulated monopole sources and measuring performance in both the near and far field, a comparison of popular spiral array types was performed [17] showing that the Underbrink spiral design [16] possessed the best all-round performance in the acoustic near and far field. Small improvements in sidelobe characteristics have been achieved by modifying a typical logarithmic array pattern by focussing more of the microphones near the array centre while still spanning the same array area [18, 19] for sources that are located near the centre of the scanning grid [17]. Efforts have been made to optimise array designs by using cost functions to minimise MSL [20–23] that yield significant improvements of MSL relative to an initial array design yet none of these designs conclusively display a superior combination of MSL and MLW compared to logarithmic spiral arrays and their variations. A unique algorithm is presented in this paper that iteratively removes microphones from an existing array that is significantly larger than the desired array based on sidelobe and main lobe criteria. This concept was inspired by a Finite Element Analysis study conducted by Xie and Steven [24]. They removed redundant material from a structure under various loading conditions by selecting the elements that possessed the least amount of von-Mises stress. Via iteration and using various initial geometries, they arrived at optimised structures that possessed the least amount of elements to meet allowable stress and strain criteria. This concept is analogously applied here; the large initial array of microphones represents the structural elements that are reduced to a desired number of microphones and the product of frequency-averaged MSL and MLW represents the structural criteria. In this study, the microphone that produces the smallest product of frequency-averaged MSL and MLW of the beamformer response when shaded “off’ is detected using CSB. This microphone is then removed and this process is repeated until the predetermined desired number of microphones is reached. This method was used to design several 48-channel microphone arrays from 841 and 961-channel grid array stencils, over a frequency range of 2000 Hz to 8000 Hz, to meet typical small scale anechoic wind tunnel environments. These arrays possess superior products of frequency-averaged MSL and MLW over the considered logarithmic spiral and randomised pattern arrays in this paper. A secondary implication of this algorithm is being able to determine the least effective microphones within an existing array. For example: if some microphones become faulty within an array and tests must proceed prior to their replacement, this method can be used to determine which microphones should be removed based on a beamforming-frequency range of interest.
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III. Beamforming Methods
A spherical wave source (i.e., a monopole source) of unit source strength in still conditions is simulated. The propagation of this wave to the array plane can be represented as p, a vector of complex pressures (Pa) in the frequency domain (M × 1 vector), defined as p( f , m) =
1 − j2π f rs (m) exp [ ] 4πrs (m) c
(1)
where f represents frequency (Hz), c is the speed of sound in air (343 m/s) and the variable m denotes a microphone number and ranges from 1 to M, where M is the number of microphones in the array. The distances from the source point to the microphones are given by rs (m). The cross-spectral beamforming algorithm [14] requires the measurement of complex pressures at each microphone. The pressures generated by Eq. (1) are processed in a cross-spectral matrix, C, which is an M × M matrix defined as C = ppT
(2)
where T represents the complex transpose and conjugate. The diagonal entries are set to zero to remove the autospectra from the matrix [14]. A beamforming output is computed over a square-planar discretised grid of N data points (scanning grid) at a known ̂, contain the distance from the array, positioned in line with the centre of the microphone array. Steering vectors, υ unique distances of each scanning grid point to each microphone, m. The steering vector for the mth microphone is a N × 1 array defined as υ( f , m) =
1 − j2π f rm (m) exp [ ] 4πrm (m) c
̂ = [υ(1), υ(2)⋯υ(M)] υ
(3) (4)
where rm is the vector between the scanning grid point to the microphone m. The cross-spectral beamforming output, Y , [1] is computed using Y (̂ υ) =
̂T {wCwT } υ ̂ υ 2
M M (∑m=1 w(m)) − (∑m=1 w(m))
(5)
where w represents the 1 × M microphone shading vector. The shading quantity adjusts the microphone pressures relative to each other. In this study, values of w(m) are either 0 or 1 to simulate a microphone being removed or included in the array respectively. It can be easily seen by comparing Eqs. (1) and (3) that if rm (m) = rs (m), then Eqs. (1) and (3) will be equal, which is the detection of the source. However, for some scanning grid points such that rm (m) ≠ rs (m), a non-zero array response is produced, namely sidelobes. The characteristic performance of the array can be determined by the ratio of the main lobe amplitude to the amplitude of the biggest sidelobe; i.e., the MSL. This array response to a monopole source of unity strength located at the centre of the scanning grid (x = y = 0) is called the Point Spread Function (PSF) [14]. To determine the PSF, a suitable scanning grid area (∣x∣ × ∣y∣) and the distance of the array to the scanning grid plane (z) must be defined. Note that in this study only square-sized scanning grids were considered and therefore ∣x∣ is implied to be equal to ∣y∣. These parameters are identified in Fig. 1. In this study, a single monopole source is considered at the scanning grid centre (x = y = 0) and z = 600 mm, based on previous experimental conditions in the Anechoic Wind Tunnel (AWT) at The University of Adelaide [4]. Many scanning grid areas are investigated in this paper, similarly based on previous research in the AWT [4–6, 18] and others [1, 20, 25]: ∣x∣ × ∣y∣ = 500 mm × 500 mm to 3000 mm × 3000 mm. Each scanning grid consists of 100 × 100 lines, such that there exists N = 101 × 101 = 10,201 scanning grid points. Typically such a finely spaced scanning grid is impractical for aeroacoustic applications, especially when the data are further post-processed by potentially computationally expensive algorithms such as DAMAS and CLEAN-variations that are dependent on N. However, in order to minimise discretisation errors when comparing array types, the MSL and MLW must be calculated as accurately as possible within reasonable and practical limitations.
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Array plane
| z | = 600 mm Steering vectors
Monopole source at x=y=0
Example scanning grid point Scanning grid area: |x| |y|
|y| |x|
Fig. 1 Schematic diagram showing the array plane, scanning grid plane and representative steering vectors for the monopole source and an example scanning grid location.
IV. Array Reduction Method
The beamformer response to a PSF, Y , is computed at fs frequencies between fmin and fmax over a scanning grid of ∣x∣ × ∣y∣ comprised of N scanning grid points. In this study, fmin = 2000 Hz, fmax = 8000 Hz, fs = 10 (equispaced between fmin and fmax ) and ∣x∣ × ∣y∣ = 1000 mm × 1000 mm. Note that the array reduction method in this paper develops an array using a 1000 mm × 1000 mm scanning grid comprised of N = 51 × 51 = 2601 points, yet the array’s performance Y is computed and analysed over a wider variety of scanning grid areas with N = 101 × 101 = 10,201 scanning grid points. The coarser scanning grid resolution is used to significantly reduce the overall array reduction computation time. At each frequency fs , the MLW is calculated first by determining how many scanning grid points the main lobe covers until it decreases by 3 dB from the main lobe peak, consistent with others [1, 22, 26]. A larger region calculated around the main lobe named the Main Lobe Area (MLA) is defined here by 30 dB below the normalised main lobe peak (shown in Fig. 2a). Due to the potential asymmetry of the main lobe, the bounds of the MLA are calculated in both the x and y directions, namely MLA (x) and MLA (y) respectively, as shown in Fig. 2b (i.e., MLA = MLA (x) × MLA (y)). For clarification, the MLW is also calculated in both the x and y directions and the maximum of the two is defined as the MLW. The MLA is removed from the image source map so that the remaining source map does not contain the main lobe to within 30 dB of the main lobe peak. The MSL is then evaluated as the maximum remaining CSB value. Note that if the MLA is not removed to within 30 dB relative to the main lobe, then the MSL may be inaccurately calculated as part of the lower-slope of the main lobe. The initial array stencils used in the array reduction method are described first, to illustrate the initial conditions of the simulations. This is followed by the two array reduction methods. The first method involves removing the microphones one at a time, while the second method involves removing multiple microphones at a time where the number of microphones removed decreases per iteration (following an exponential decay profile). The array output performance parameters are then introduced. A. Initial Array Stencils An initial array stencil is required for the array reduction algorithm such that there are sufficient channels for the algorithm to produce meaningful results and that there are no discernible patterns of microphones that may bias the final reduced array pattern. Such bias nonetheless could be beneficial in alternative array designs for specific applications. 4
Main lobe
Main lobe amplitude : 0 dB Main lobe width cut-off : -3 dB
MLW Main lobe cut-off : -30 dB Sidelobe calculation area
Sidelobe calculation area
Main lobe area (MLA)
(a)
(b)
Fig. 2 (a) Schematic diagram illustrating MLW and the region that encompasses the main lobe, main lobe area (MLA). (b) Example map of Y obtained using CSB, illustrating the differences between MLA (x), MLA (y) and MLW. The MSL is also identified. The initial array stencils considered in this study are presented in Fig. 3 and are designed with the following two criteria; the minimum spacing between the microphone centres, δmin , must be no less than 20 mm to simulate real-world microphone installations within an array and the microphones must fit within a 1000 mm × 1000 mm area. To compactly fit a large number of microphones in a confined area leads to the obvious choice of using a finely spaced uniform grid array. Equispaced grid arrays of 961 and 841-channels are used as initial array stencils and are shown in Figs. 3a and 3b respectively. The minimum spacings between the microphones, δmin , are approximately 33 and 36 mm respectively. To simulate finer microphone spacing in critical areas, such as near the centre and avoiding the diagonal quadrants (see Fig. 7), non-equispaced grid patterns are also used as initial array stencils. To ensure that microphones are well populated near the centre of the array and are more sparsely distributed near the outer regions of the array, the spacing between the microphones are calculated using a power-relationship with respect to linear spacing. A linearly spaced vector is generated from 0 to 1, with n points. Each row of the initial array stencil possesses 2n – 1 points, thus a total number of (2n − 1)2 microphones exist in the array. For example, if n = 15, there are 2 × 15 – 1 = 29 microphones in each row and thus a total of 292 = 841 microphones. The microphone coordinates of this linearly spaced array are raised to a power and then multiplied by half the total array size (calculated as 500 mm in this study). The power to which the linearly spaced values are raised to is dependent on n. If n is large, a high power will cause δmin < 20 mm near the array centre, which is unacceptable. If the power is low (near unity) with large n, the points are nearly equispaced. Figure 3c displays such an array, using n = 15 and a power of 1.2, yielding δmin ≈ 21 mm. A randomised initial stencil pattern would be advantageous as the beamformer output could possess weaker grating lobe distributions, yet this presents a very difficult challenge. Such arrays are able to be produced up to only 169-channels due to the immense number of permutations of array spacings that significantly decrease the statistical likelihood of randomly filling a 1000 mm × 1000 mm area without violating the δmin criteria. B. One-by-one Microphone Removal An initial array stencil is used that is composed of a much larger number of microphones, Mi , relative to the desired number of microphones, M f . From this stencil one microphone at a time is systematically removed such that this reduced array results in the smallest product of the frequency-averaged MSL and frequency-averaged MLW. To achieve this, every microphone from m = 1:Mi is shaded one-at-a-time using w. Let m′ be the microphone that is shaded, so that w(m′ ) = 0 and w(m) = 1, for m = 1 ∶ Mi where m ≠ m′ . The beamformer output Y is calculated for every m′ value of m in 1:Mi . The microphone number m′ that produces the minimum frequency-averaged product of MSL and MLW will be removed from the array, because by removing this microphone, the product of MSL and MLW decreases the least (i.e., removing this microphone causes the smallest negative impact on Y ). The size of the array then decreases by one and the next shading process commences: w(m′ ) = 0 and w(m) = 1, for m = 1 ∶ Mi − 1 where m ≠ m′ . This process is 5
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Fig. 3 Initial array stencils, using grid-patterns. Each arrays is confined within a 1000 mm × 1000 mm area. (a) Equispaced 961-channel, (b) Equispaced 841-channel and (c) Non-equispaced 841-channel. repeated until the predetermined desired number of microphones remains within the array stencil, M f , corresponding to 48-channels in this study. The number of matrix multiplications involved in producing Y shown in Eq. (5) are proportional to N, fs and M 4 . Therefore the computational time to perform the one-by-one microphone removal rapidly increases with larger values of M f . A desktop PC with an Intel(R) Xeon(R) CPU E5-2630 v4 dual-core 2.20 GHz processor with 64.0 GB of RAM was used for these MATLAB-compiled simulations. Using N = 51 × 51 = 2,601 and fs = 10, solution times of approximately 1 hour for M f = 121, 11 days for M f = 841 and 18 days for M f = 961 were recorded. Based on these solution times and using the same values of N and fs , an expected solution time for M f = 2601 (being an equispaced δ = 20 mm array stencil over an 1000 mm × 1000 mm area) would be in the order of 5 to 10 years. C. Exponential Decay Profile Removal Due to the significant computational expense of removing one microphone at a time from a large initial array, a modification to the removal method is proposed to markedly improve the computational speed while attempting to produce the best quality reduced array possible. An Exponential Decay Profile (EDP) function, E, is introduced that dictates the number of microphones to be removed per step E (γ, β) = Mi γe−β m⃗ + 1
(6)
where γ and β are adjustable parameters that determine the number of microphones removed at the first iteration and the ⃗ is a linearly spaced vector from 1 to Mi − M f . Note that if steepness of the exponential decay profile respectively and m γ → 0 and β → ∞, then EDP → 1 which would remove all of the microphones one-by-one. Each of these parameters are investigated in this study, listed in Table 1 and graphically represented in Fig. 4. The solution times of each EDP compared to the one-by-one removal method are also listed. It can be seen that using an EDP can produce a reduced array approximately ten-to-twenty times faster than the one-by-one removal method. The parameter γ determines how many microphones are removed at the first iteration. An initial estimate set γ to a value that would result in approximately 10% of the initial array to be removed in the first iteration. By using a much larger value of γ it is possible that an excessive number of microphones could be removed due to the specific sidelobe and main lobe characteristics of the initial array stencil, which would not be indicative of the array output characteristics at further stages of reduction (i.e., microphones that are of little importance at the first iteration step may be useful in subsequent stages). Using a much smaller of γ results in little time saving compared to the one-by-one removal process. It is important that the microphones be removed one-by-one during the final stages of the array reduction process. This can be controlled by adjusting β. Removing more than one microphone at a time towards the end of the simulations (where M ≤ 150 to 200) produces M f = 48 channel arrays with poorer MSL and MLW characteristics than those using one-by-one microphone removal. In contrast, EDP reduced arrays that are generated by removing microphones one-by-one for a substantial part of the removal process (M ≈ 300 − 400) show no significant improvement in MSL and MLW characteristics. Therefore, as one may expect, there exists an optimum balance between computation time-saving and quality of reduced array outputs. In Table 1, the parameter α represents how many microphones are removed
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Table 1 Exponential Decay Profile (EDP) parameters for Mi = 961, M f = 48. The solution times are recorded using fs = 10 and N = 2601. The Mi = 961 one-by-one simulation data (ONE) are included for reference. Name
γ
EDP1 EDP2 EDP3 EDP4 ONE
10 10 10 12 -
First Removal No. Mics. 85 84 82 99 1
β
% of Mi 8.8 8.7 8.5 10.3 0.1
13 15 17 16.5 -
Solution time
α Total 184 286 366 240 961
% of Mi 19.1 29.8 38.1 25.0 100
(hours) 18 31 48 24 423
Number of microphones removed per step
one-by-one to complete the simulation. For example, if Mi = 961, M f = 48 and α = 286, then microphones m = 334 (= 286 + 48) to 49 are removed one-by-one.
100 80 60 40 20
Fig. 4
0 961 900
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Exponential Decay Profiles (EDP) using the parameters listed in Table 1.
D. Array Evaluation Method To compare the reduced arrays with each other and with other array designs, both MSL and MLW are used and defined within this paper as MSL(dB) = 20 log10 {
Ys } Ymax
(7)
N3dB } (8) N where Ys is the maximum sidelobe pressure, Ymax is the non-normalised main lobe pressure and N3dB is the number of scanning grid points that are occupied by the main lobe from its normalised maximum amplitude (0 dB) to -3 dB. Note that MLW is also defined as the physical size of the main lobe, 3 dB down from its peak and is listed as MLW (mm) (as defined in Section IV). Both values are presented in this paper. For clarity, the MLW definition in Eq. (8) in figures will be simply written as N3dB /N (%) and the physical distance quantity as MLW (mm). A dimensionless metric, Φ, is introduced that combines MSL and MLW into a single value that characterises an array. This is the formal definition of the parameter used to determine the microphone to be removed during the array reduction process. It is also a convenient way to compare beamforming arrays with a single value, at each beamforming frequency and scanning grid size. This is simply MLW(%) = 100 × {
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Φ = 10 log10 {
Ys N3dB × } + 20 Ymax N
(9)
The metric Φ can also be averaged with respect to frequency to produce Φav , which can then be evaluated over a range of scanning grids (500 mm × 500 mm to 3000 mm × 3000 mm) ⎧ ⎫ ⎪ N3dB ⎪ ⎪ 1 fs Ys ⎪ Φav = 10 log10 ⎨ ∑ ( × )⎬ + 20 ⎪ f Ymax N ⎪ ⎪ ⎪ ⎩ s 1 ⎭
(10)
V. Popular Arrays for Comparison
For comparison against the results of the array reduction method, two types of arrays are considered: logarithmic spirals and randomised pattern arrays. Each of these array types are confined to within a 1000 mm × 1000 mm area, M = 48-channels and δmin = 20 mm. To help compare the distributions of the microphones of the two array types, an algorithm is written to determine δ of the logarithmic and randomised arrays. The x and y differences of the microphone locations within the array are calculated, squared, summed together and square-rooted to arrive at the distances between each microphone. For an array of M-microphones, there exist M × (M − 1)/2 values of δ. The average of these values is δav and the minimum is δmin . A. Logarithmic Spiral Logarithmic spirals are investigated here as they possess the highest number of unique spacing between microphones as compared to other array types of the same size and number of microphones [16]. The parametric equations for the coordinates (x and y) of a logarithmic spiral are x(m) = a cos [θ(m)] ebθ(m)
(11)
y(m) = a sin [θ(m)] ebθ(m)
(12)
where a and b are coefficients that affect the overall size of the spiral and how rapidly the arms of the spiral expand from the centre respectively. The angle θ(m) is defined as (m − 1)ϕl (13) M ϕ is the spiral sweep angle (ϕ = 2π represents one spiral revolution and is constant for all of the logarithmic pattern spiral arrays in this paper) and l represents a multiple of ϕ. Table 2 lists these quantities used in this comparative study and Fig. 5 displays the resulting array patterns. From Table 2 it is deduced there are no clear correlations between δmin , δav and δmax , ensuring that a fair, unbiased range of logarithmic spiral arrays are produced for comparison with other array types. It can be observed in Fig. 5 that each of the logarithmic arrays are geometrically distinct. Maps of Y calculated from the PSFs of the logarithmic spiral arrays are presented in Fig. 5. These are evaluated using the conditions shown in Section IV and are post-processed to calculate MSL, MLW and Φ, as presented in Fig. 5. By observation of Figs. 5f and 5g, Log3 and Log4 possess the smallest MLW on average across the frequency range and Log5 the largest. The results in Fig. 5h illustrate the strong variance of MSL with respect to frequency. Log1, Log4 and Log5 have relatively low MSL values at low frequencies yet their high frequency MSL values are much greater. Both Log2 and Log3 have less MSL variance with respect to frequency, with Log2 possessing the smallest averaged MSL value across this frequency range. The values of Φ for these arrays are presented in Fig. 5i, showing that Log2 is the superior logarithmic array within this frequency range over a scanning grid area ∣x∣ × ∣y∣ = 1000 mm × 1000 mm. This is determined by calculating Φav over this frequency range and observing that Log2 possesses the lowest value for the logarithmic arrays considered here. Log2 does not possess the lowest frequency-averaged MSL or MLW, yet its overall performance combining these two parameters is recognised by using the parameter Φ. Thus Log2 is the logarithmic spiral array that will be used for comparison with the arrays generated by the reduction process. θ(m) =
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Fig. 5 Logarithmic spiral pattern arrays for comparison with the array reduction method and the analysis of their beamformer output. Arrays contained in figures (a) through (e) are labelled Log1 through Log5 and are each contained within a 1000 mm × 1000 mm area. The beamformer output of the PSF is conducted over a ∣x∣ × ∣y∣ = 1000 mm × 1000 mm scanning grid placed ∣z∣ = 600 mm from the array resulting in (f) MLW (mm), (g) N3dB /N (%), (h) MSL (dB) and (i) Φ (dB).
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Table 2 Parameters used in Eqs. (11) and (12) to create the patterns and microphone spacings, δ, for the logarithmic spiral arrays shown in Figs. 5a to 5e. For all arrays, ϕ = 2π. Name Log1 Log2 Log3 Log4 Log5
a 37 70 95 97 38
b 42 70 30 20 66
l 11 5 10 15 20/3
δmin (mm) 21.8 46.2 34.3 41.9 20.5
δav (mm) 335.4 366.9 397.6 406.1 303.0
δmax (mm) 1223.1 1080.1 1118.3 1097.7 1013.0
Table 3 Methods used to create the patterns and microphone spacings, δ, for the randomised array patterns shown in Figs. 6a to 6e. Name Rand1 Rand2 Rand3 Rand4 Rand5
Method Grid Non-Grid Grid Non-Grid Non-Grid
δmin (mm) 20.8 29.0 26.3 22.0 29.4
δav (mm) 506.5 544.3 527.3 528.8 500.3
δmax (mm) 1233.4 1191.6 1266.0 1188.2 1243.0
B. Randomised Pattern Two methods are used here to generate randomly spaced arrays. The first method, namely the “Non-Grid” method, iteratively develops x and y coordinates based on random numbers using a minimum spacing constraint of δmin = 20 mm within a 1000 mm × 1000 mm area. The array coordinates can be separated by 20 mm in any direction, not just the x and y directions. The second method, namely the “Grid” method, guarantees a minimum spacing of 20 mm for every array designed. A 2601-channel equispaced grid array with δav = δmin = 20 mm is first created spanning 1000 mm × 1000 mm and then its entries are randomly populated with the desired number of microphones (48-microphones in this paper). This method is much faster than the Non-Grid method for arrays much larger than 100-channels, but the array coordinates using the Grid method are bound by the grid spacing. Within each randomised array generating algorithm, the MSL of each array is calculated in the same manner as detailed in Section IV. If the frequency-averaged MSL of the array is greater than the previous array, it is discarded and the lowest-MSL array is retained. This iterative search was conducted over one-million array permutations in attempt to produce the best performing randomised array possible and thus a fair comparison with other arrays. The mean and minimum microphone spacings achieved using both methods are presented in Table 3 and the resulting array patterns are presented in Fig. 6. It is important to produce randomly spaced arrays with a variety of δ values, yet this is very difficult to control. From Table 3, δmin ≈ 20 mm to 30 mm and both δav and δmax are similar for all of the arrays, with mean values of approximately 520 mm and 1220 mm respectively. Compared to the logarithmic spiral δ values presented in Table 2, the randomised arrays have less variation in δ values between them. The randomised array values of δav and δmax are also greater than the logarithmic spiral arrays. The randomised arrays presented in Fig. 6 are compared with each other using the same method as the logarithmic spiral arrays. As observed in Figs. 6f and 6g, the MLW values of the randomised arrays are similar and all are considerably smaller than the logarithmic arrays. On average, Rand2 displays the smallest MLW. Conversely, the randomised arrays present larger MSL values than the logarithmic arrays, which can be seen by comparing Figs. 5h and 6h. The best performing array based on frequency-averaged MSL is Rand1, which presents relatively constant MSL values across the frequency range. As observed in Fig. 6i, Rand1 also possess the lowest value of Φav and is therefore used for comparative analysis with the arrays generated by the array reduction method. The logarithmic spirals display lower values of Φ for approximately f ≤ 4000 Hz than the randomised arrays, yet their values of Φ at 4000 Hz ≤ f ≤ 8000 Hz are comparable.
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Fig. 6 Randomised pattern arrays for comparison with the array reduction method and the analysis of their beamformer output. Arrays contained in figures (a) through (e) are labelled Rand1 through Rand5 and are each contained within a 1000 mm × 1000 mm area. The beamformer output of the PSF is conducted over a ∣x∣ × ∣y∣ = 1000 mm × 1000 mm scanning grid placed ∣z∣ = 600 mm from the array, resulting in (f) MLW (mm), (g) N3dB /N (%), (h) MSL (dB) and (i) Φ (dB).
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Table 4 Methods used to create the patterns and microphone spacings, δ, for the reduced array patterns shown in Figs. 8a to 8d. Name 841NonEq-ONE 961Eq-EDP2 961NonEq-EDP2 961Eq-ONE
δmin (mm) 38.7 33.3 20.2 33.3
δav (mm) 344.6 382.6 414.3 407.7
δmax (mm) 913.9 1046.1 1004.2 1020.3
VI. Results
An Mi = 961-channel array shown in Fig. 3a is used as the initial array stencil for a one-by-one array reduction process, where fmin = 2000 Hz, fmax = 8000 Hz and fs = 10. This array is reduced to an Mi = 48-channel array, shown in Fig. 7e. Progressive development of the array reduction and the evolution of the MSL can be viewed in Fig. 7. This reduction process is typical for the array reduction methods considered in this paper. The EDP reductions require far fewer reduction stages, yet their array patterns at each stage are very similar. It can be observed in Fig. 7a that the microphones from the array corners and extending along the diagonals toward the centre are removed first, producing a cross-like pattern. This results in an a decrease of MSL to below -30 dB upon removal of the first approximate 100-channels. Some redundant microphones in a densely spaced grid array often contribute to strong grating lobes at 90° angles to the main lobe centre [19, 23] and their removal decreases the strength of the grating lobes. Progressing to Figs. 7b and 7c, there do not appear to be any clear discernible patterns of array removal and only small changes in MSL, all of which are less than -30 dB. At M ≈ 140, the sidelobe magnitudes increase to above -30 dB, commencing at f = 3000 Hz and f ≥ 6000 Hz. At M ≈ 80, sidelobes above -30 dB emerge for f = 4000 Hz and 5000 Hz. The MSL increases rapidly from M ≈ 100 corresponding to 2 ×M f (Fig. 7d) to M f = 48 (Fig. 7e). Every microphone removed in this stage is critical and has a significant effect on the MSL. The MSLs at f = 4000 Hz, 6000 Hz and 8000 Hz become the strongest. Interestingly, the MSL at f = 3000 Hz, which emerges first during the array reduction process, possesses the strongest MSL at M ≈ 53, then decreases with further array reduction. This frequency is the only exception to the general trend of increasing MSL with increasing number of removed microphones. The frequency-averaged MLW has small fluctuations from Mi = 961 to M ≈ 140, as observed in Fig. 7g. This was indicative of all fs = 10 frequencies analysed. Therefore, within this range the reduction method is predominantly MSL-based. From M ≈ 140 to M f = 48, the MLW increases rapidly and fluctuates to-and-from its maximum value and the array reduction method has stronger influences on both MSL and MLW. The array reduction processes at M ≈ 140 and 80 appear to be critical points for both the MSL and MLW, showing sudden increases in both MSL and MLW. A. Reduced array patterns The four-reduced arrays with the best values and distributions of Φ with respect to frequency are presented in Figs. 8a through 8d, named 841NonEq-ONE, 961Eq-EDP2, 961NonEq-EDP2 and 961Eq-ONE respectively. The name of each array describes the array reduction method required to generate it. For example, 961NonEq-EDP2 means that a non-equispaced Mi = 961 initial array is used and reduced using an EDP2 method, and 961Eq-ONE means an equispaced Mi = 961 initial array is used and reduced one-by-one. By visual observation, each of these arrays possess very few microphones near the array area corners. Arrays 841NonEq-ONE and 961Eq-EDP2 show a majority of microphone population within the centre of the array area. An interesting observation of 961Eq-EDP2 are the arms that appear to extend from the centre population of the array, similar to a logarithmic spiral arm. Arrays 961NonEq-EDP2 and 961Eq-ONE have a similar shape, with 961Eq-ONE having a slightly more even distribution of microphones. Despite these few similarities, the arrays are quite distinct. The values of δ for the reduced arrays are presented in Table 4. All of the reduced arrays possess smaller δmax values than both the logarithmic spiral and randomised arrays. The δav values are very similar to the logarithmic spiral arrays and smaller than the randomised pattern arrays. The δmin values for the arrays derived from an Mi = 961-channel equispaced grid array are restricted to a minimum value of 33.3 mm, which is observed in Table 4 for both the one-by-one and EDP2 arrays. By comparing all of the δ values of Log2 with the reduced array δ values, there are no clear correlations to discern which δ values produce a better performing array in terms of MSL and MLW. Using EDP2 produces superior results compared to the other EDP profiles and is the only EDP considered further in this paper. The frequency-averaged MSL of the EDP2 reduced array is lower than the other EDP reduced arrays by 2 dB 12
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or more, while also possessing the smallest frequency-averaged MLW. Whether an equispaced or non-equispaced initial array stencil is used produces no conclusive result to suggest one initial array stencil is superior to the other. On average, using Mi = 961 produces arrays with lower Φ values than those generated using Mi = 841. Preliminary simulations using smaller initial arrays stencils of Mi = 121, 169 and 225-channels lead to reduced arrays with poorer MSL and MLW values. Therefore, the higher the value of Mi , the better the reduced array, yet the optimal distribution of microphones within the array stencil is still unclear. Other computationally expensive preliminary simulations, conducted with Mi = 2601 and using very aggressive EDP (i.e., γ > 15%) do not produce a reduced array with improved Φ relative to the arrays presented in Fig. 8. By collating these observations it is proposed that using one-by-one array reduction with as large an initial array stencil as possible produces the best reduced array, yet with such massive increases in computational requirement, the small improvements in MSL and MLW may be heavily outweighed by the exceptionally long computation times. The array possessing the smallest MLW is 961Eq-ONE, as observed in both Figs. 8e and 8f. The worst performing array in terms of MLW is 841NonEq-ONE, over the range 2000 Hz ≤ f ≤ 6000 Hz. The differences in MSL values over the analysed frequency range is much more significant than the MLW values. As observed in Fig. 8g for approximately f ≤ 4700 Hz, 961Eq-EDP2 has superior MSL values with a minimum of approximately -29 dB. The 841NonEq-ONE array also performs well in this frequency range. The 961NonEq-EDP2 possesses a relatively flat response across the frequency range, with a gradual increase in MSL with respect to frequency. The 961Eq-ONE array shows a local maximum at f ≈ 3200 Hz and the MSL values also gradually increase with respect to frequency. At f ≥ 6000 Hz, no array clearly outperforms the other in terms of MSL. The values of Φ in Fig. 8h reveal the overall array performance more clearly. The 961Eq-EDP2 array possesses the lowest values of Φ for f ≤ 4500 Hz, yet also has the highest values within 4500 Hz ≤ f ≤ 6000 Hz and 6900 Hz ≤ f ≤ 7500 Hz. All of 841NonEq-ONE, 961NonEq-EDP2 and 961Eq-ONE perform relatively consistently with respect to frequency, in that they do not display any strong local minima or maxima. This is a desirable quantity, in that for any particular beamforming frequency of investigation, a similar performance within a frequency band of approximately 1000 Hz can be expected from one frequency to another. Such a characteristic is not observed in both the logarithmic spiral and randomised pattern arrays, by inspection of Figs. 5i and 6i respectively. A series of frequency-averaged calculations of MSL, MLW and Φ, using a range of scanning grid sizes, are presented in Fig. 9. The order of MLW performance between each the arrays is generally constant with each scanning grid size (i.e., for each scanning grid size, the MLW of Rand1 outperforms all arrays and Log2 has the worst MLW performance). As observed in Figs. 9a and 9b, the 961Eq-ONE array has the lowest averaged-MLW value of the reduced arrays. The MSL values between the different arrays have the greatest disparity at the smaller scanning grid areas as seen in Fig. 9c. At ∣x∣ = ∣y∣ = 1000 mm, corresponding to the scanning grid size used for the array reduction process, 841NonEq-ONE has the lowest MSL value which is very slightly less than the Log2 MSL value. The Rand1 array has the largest MSL value and the other reduced arrays are within 1 dB of each other. For ∣x∣ ≥ 1500 mm, the averaged-MSL values of each array converge to each other, showing a decreasing disparity in averaged-MSL with increasing scanning grid area. The overall performance of the array is represented by Φav which is presented in Fig. 9d. At ∣x∣ = ∣y∣ = 1000 mm the data is clustered around similar Φav values. The 961Eq-ONE array possesses the lowest value of Φav ≈ -17.4 dB. The 961Eq-EDp2 and 961NonEq-EDp2 arrays have similar values of Φav ≈ -17.2 dB. The other arrays have Φav > -17 dB, with Log2 having the highest value of -16.7 dB. The smallest Φav values are obtained using Log2 for ∣x∣ = ∣y∣ = 700 mm and 800 mm and Rand1 for ∣x∣ = ∣y∣ ≥ 1200 mm. This shows that the array reduction method acts very specifically for the specified scanning grid range used during the simulations. The difference in Φav between 961Eq-ONE and the EDP2 arrays (961Eq-EDP2 and 961NonEq-EDP2) is approximately 0.2 dB. Based on this value alone, the considerable difference in computational effort to obtain 961Eq-ONE compared to the EDP2 arrays seems to yield little significant benefit. This however may not occur for different initial and final array sizes and frequency ranges and increments. Thus, from this present study, it is concluded that using the one-by-one method yields a lower Φav value at the scanning grid range used during the array reduction process; knowing how much it surpasses the best EDP array for generalised conditions will be considered for future work.
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VII. Conclusions
A unique array design method is presented in this paper that involves an array reduction method. From a large initial array stencil, Mi = 841 and 961, microphones were removed based on the minimum frequency-averaged product of the MSL and MLW. An overall beamforming array performance metric was defined by combining the MSL and MLW performance into a single quantity, Φ. Simulations were conducted over a frequency range of fmin = 2000 Hz to fmax = 8000 Hz in fs = 10 frequency increments and using an array size covering a 1000 mm × 1000 mm area and over a scanning grid of the same size with 2 N = 2601 points. 1 From the initial array, Φ is evaluated. Each microphone is shaded off, one at a time. The microphone when shaded off, that produces the least-increase of Φ, is removed, and the process is repeated until M f = 48 microphones remain. Microphones were removed in two ways: one-by-one, or following an EDP. The one-by-one method has significant computational cost whereas the EDP method is ten-to-twenty times faster. At the designated scanning grid size of 1000 mm × 1000 mm, the array generated by the one-by-one method shows a smaller Φav compared to the other EDP generated arrays. The improvement relative to the increase in computation time is subjective. 16
Reduced arrays generated by both methods show improvements compared to the logarithmic spiral and randomised pattern arrays presented in this paper, in terms of Φ and almost in all cases, MLW. In some cases, the logarithmic spiral possessed smaller MSL values compared to the reduced arrays yet their value of Φ was always greater over a 1000 mm × 1000 mm scanning grid. The best logarithmic spiral array and randomised pattern array were compared against the best reduced arrays, over a range of scanning grid sizes, from 500 mm × 500 mm to 3000 mm × 3000 mm. While the reduced arrays were generated using a 1000 mm × 1000 mm scanning grid, they also show superior values of Φav over logarithmic arrays across the majority of the different scanning grid sizes. The MSL and Φ variance with respect to frequency is less than the logarithmic spiral and randomised pattern array variance, implying that the reduced arrays have a more consistent response across the frequency range of interest. This array reduction method leads to many practical advantages. A frequency range specific to an acoustic test can be used to design the array. The initial array stencil can be varied to suit the overall array size and enforce the locations and density of microphones in specific locations. The array reduction method can be used over any scanning grid size, that sufficiently captures the main lobe area. Furthermore, the array can be fitted to a grid, making the exact locations of microphones easier to quantify and more importantly, the array microphone locations can be easily interchanged within a grid structure inside a wind tunnel or on a moveable-wall mounting to optimise the MSL and MLW characteristics for a specific acoustic test. This is also therefore very suitable for a small-scale facility with a restricted budget, to achieve high quality beamformer outputs within specific frequency bands. In addition, if a microphone becomes faulty in an array, the reduction algorithm can be used quickly to determine which microphone position should be discarded from the array pattern to have the least negative impact on MSL and MLW in the beamformer output. Future research will include using the array reduction method at single frequencies, to produce array designs that are optimised for narrow frequency bands. By using a small initial stencil (approximately 169-channels), this can be a feasible practical solution for a small-scale anechoic wind tunnel, where a 48-channel array can be modified to suit the beamforming frequencies of interest for each experiment. Using different EDPs and significantly faster Graphics Processing Unit (GPU) computation, finer initial array stencil grid spacings of various geometries and simulations with higher fs and N can be used to arrive at superior M f -channel array designs. Deeper research could also be conducted to determine the realistically-attainable value of Φmin for a specific frequency range, array and scanning grid size. These concepts are currently in a preliminary research stage.
Acknowledgements
This research was supported by the National Natural Science Foundation of China (Grant No. 11772146) and the Science and Technology Innovation Council of Shenzhen (Grant No. JCYJ20170817110605193).
References
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