Sparsity constrained deconvolution approaches for acoustic source mapping Tarik Yardibi and Jian Lia兲 Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida, 32611
Petre Stoica Department of Information Technology, Uppsala University, P.O. Box 337, SE-751 05 Uppsala, Sweden
Louis N. Cattafesta III Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, Florida, 32611
共Received 2 November 2007; revised 11 February 2008; accepted 14 February 2008兲 Using microphone arrays for estimating source locations and strengths has become common practice in aeroacoustic applications. The classical delay-and-sum approach suffers from low resolution and high sidelobes and the resulting beamforming maps are difficult to interpret. The deconvolution approach for the mapping of acoustic sources 共DAMAS兲 deconvolution algorithm recovers the actual source levels from the contaminated delay-and-sum results by defining an inverse problem that can be represented as a linear system of equations. In this paper, the deconvolution problem is carried onto the sparse signal representation area and a sparsity constrained deconvolution approach 共SC-DAMAS兲 is presented for solving the DAMAS inverse problem. A sparsity preserving covariance matrix fitting approach 共CMF兲 is also presented to overcome the drawbacks of the DAMAS inverse problem. The proposed algorithms are convex optimization problems. Our simulations show that CMF and SC-DAMAS outperform DAMAS and as the noise in the measurements increases, CMF works better than both DAMAS and SC-DAMAS. It is observed that the proposed algorithms converge faster than DAMAS. A modification to SC-DAMAS is also provided which makes it significantly faster than DAMAS and CMF. For the correlated source case, the CMF-C algorithm is proposed and compared with DAMAS-C. Improvements in performance are obtained similar to the uncorrelated case. © 2008 Acoustical Society of America. 关DOI: 10.1121/1.2896754兴 PACS number共s兲: 43.60.Fg, 43.60.Jn, 43.60.Pt, 43.60.Mn 关EJS兴
I. INTRODUCTION
Microphone arrays have become an important element of aeroacoustic testing in recent years. These arrays can be electronically steered into desired regions in space to create an image of acoustic sources at a given frequency. This image consists of the sound pressure levels of each location in the region of interest.1,2 Perhaps, the most commonly used beamforming algorithm in practice is the delay-and-sum beamformer,1,3 which basically sums the delayed and weighted versions of each microphone signal so that the actual source signals are reinforced and the unwanted noise signals tend to cancel. A well-known issue with the classical delay-and-sum approach is that the beamforming maps are usually contaminated by large sidelobes. Sidelobes can cause both the smearing and leakage of the sources.4 Consider a scenario with two closely spaced sources so that the response of the array to the first source does not wither away before the response to the second source starts. This will result in the smoothing, or smearing, of the spectrum in the sense that the
a兲
Author to whom correspondence should be addressed. Current address: Department of Electrical and Computer Engineering, P.O. Box 116130, University of Florida, Gainesville, FL 32611. Electronic mail:
[email protected]
J. Acoust. Soc. Am. 123 共5兲, May 2008
Pages: 2631–2642
two peaks will be merged into a single broad peak. Looking from a different perspective, a strong source can yield power at other locations, where no source is present, through the convolution with the sidelobes. This latter form of degradation is called leakage. Therefore, it is usually difficult to identify the true source locations and strengths through the beamforming map obtained via the delay-and-sum method. Another problem with this method is that the beamwidth tends to increase with decreasing frequency. Various other microphone array processing methods have been developed in order to mitigate the drawbacks of the delay-and-sum beamformer. Weighting schemes that maintain a constant beamwidth over a frequency range for the delay-and-sum beamformer have been discussed in the literature.5,6 Also, several robust adaptive beamforming techniques have been proposed as alternatives to the delay-andsum method.7–9 The latter two of these techniques8,9 are parametric approaches that require the number of sources to be known. Capon type beamformers cannot provide a sparse representation of the imaging scene and fail to work for coherent sources.7 In order to remedy the sidelobe problem of the delayand-sum method, a postprocessing technique called the deconvolution approach for the mapping of acoustic sources 共DAMAS兲 was developed.10,11 This approach is considered a breakthrough in the area of aeroacoustic noise measurement
0001-4966/2008/123共5兲/2631/12/$23.00
© 2008 Acoustical Society of America
2631
and has been used widely in practice. DAMAS solves an inverse problem using the known steering vectors to decouple the actual source powers from the array characteristics based on the incoherence assumption of the noise sources. This inverse model can be written in a linear matrix equation form which, due to the structure of the matrices, cannot be solved easily. DAMAS uses an iterative scheme for solving this inverse problem. Many other algorithms related to DAMAS have been proposed in the literature.12 LORE13 is a deconvolution method that uses a two stage process to solve the inverse problem. In the first step a method similar to DAMAS is used to reduce the problem dimensions and in the second step an optimization scheme is used to solve the reduced inverse problem. DAMAS2 and DAMAS314 offer alternatives and extensions to DAMAS by using FFT and assuming that the point spread function is shift-invariant. DAMAS-C15 extends DAMAS to the coherent sources case. Although requiring high computational resources, this approach is the first deconvolution algorithm considering correlated sources. DAMAS has also been applied to the three-dimensional source localization problem.16 In this paper, we introduce new approaches for solving the inverse problem for deconvolution via exploiting sparse signal representations. The sparse signal representation is extensively studied in many areas including statistics and signal processing for the recovery of signals consisting of only few nonzero elements with a known linear relation to the measured data. The sparse signal representation problem mainly aims to find the sparsest x such that y = Ax is satisfied. Stated more formally, the objective is to minimize 储x储0 such that y = Ax where A is known and y is measured. The problem in its original form is a combinatorial problem and has an exponential complexity making it impractical.17 Fortunately, when x is sufficiently sparse,17 储x储0 can be replaced by 储x储1 which leads to a convex optimization problem that can be solved much more easily using LASSO18 or BP.19 There are many works in the literature elaborating on this relaxation from the ᐉ0 norm to the ᐉ1 norm both in the noiseless and noisy case.17,20–22 In a nutshell, when x contains a small number of nonzero elements with respect to its size, the solutions with the ᐉ0 and ᐉ1 norms coincide under some mild conditions. Alternatively, FOCUSS23 can be used to iteratively solve the sparse problem. This method requires proper initialization of x. Nevertheless, FOCUSS algorithms have been successfully applied to brain 共EEG兲 signals and far field source localization problems. A Bayesian approach can also be used to estimate x using various prior probability distributions to enforce sparsity. Although these approaches do not require a user parameter,24–26 they still need some kind of initialization. Note that LASSO can also be thought of as a Bayesian approach assuming a Gaussian likelihood for y and a Laplacian prior for x which is known to enforce sparsity. Another algorithm, the ᐉ1-SVD algorithm,27 is proposed for estimating source locations in a manner similar to BP but for the multiple snapshot and complex case. The algorithm requires an initial estimate for the number of sources. Although this estimate does not have to be exact, a small error is required for good performance. Also, the algorithm has a user parameter whose selection is not trivial. 2632
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Note that this method is designed to work with the time signals. In aeroacoustic applications, however, huge amounts of time data are collected and it is preferred to work with the covariance matrix since it requires much less storage space and is more convenient to deal with. Finally, Fuchs28 also discusses methods for the recovery of source locations for the far-field linear array case by appending a noise basis vector to the steering matrix and using deconvolution together with sparsity. It is interesting to note that these approaches in the signal processing literature are similar to DAMAS. Fuchs also provides an extension to his method by using a uniform circular array and a sparse algorithm similar to BP.29 In this method, the user parameter is selected to be a constant, which can introduce errors. The remainder of this paper is organized as follows. In Sec. II we parametrize the aeroacoustic source localization problem. After reviewing DAMAS in Sec. III, we describe the sparsity constrained DAMAS algorithm 共SC-DAMAS兲, which is an application of LASSO to the specific inverse problem of DAMAS, in Sec. IV. In contrast to selecting a constant,29 the user parameter of the algorithm is selected by an adaptive and simple method which uses the eigenvalues of the sample covariance matrix. We also discuss a way to significantly speed up SC-DAMAS. Next in Sec. V, we propose another algorithm called the covariance matrix fitting 共CMF兲 which also exploits sparsity for deconvolution but in a different way than the DAMAS formulation. Both of the algorithms are formulated as convex optimization problems and only use the sample covariance matrix. These algorithms are guaranteed to converge to the globally optimal solution and they take less computation time than DAMAS. The CMF algorithm is more robust to noise than both DAMAS and SC-DAMAS. In Sec. VI, we extend our analysis to deal with correlated sources. We propose CMF-C, which is an extension of CMF to the correlated signals case, and compare it with DAMAS-C. The performance of the algorithms is evaluated using synthetic data for various scenarios in Sec. VIII. The simulation results show that the proposed algorithms can be directly employed in aeroacoustic applications due to their simplicity and good performance. Finally, we conclude the paper in Sec. VIII. II. PROBLEM FORMULATION
Consider the wave field generated by K near-field wideband sources located at x where x 关x1 , x2 , . . . , xK兴 and xk is the three-dimensional location of signal k, k = 1 , . . . , K. At the pre-processing stage, the time data of each microphone is divided into I segments where each segment consists of L time samples. Each segment of data is then converted into L narrow frequency bins by the L-point discrete Fourier transform 共or FFT兲. In this case, the M ⫻ 1 array output vector of an M element microphone array in the presence of additive noise can be represented as,4,30 yi共l兲 = A共x, l兲si共l兲 + ei共l兲,
i = 1, . . . ,I,
共1兲
for l = 1 , . . . , L, where A is the M ⫻ K steering matrix defined as A共x , l兲 关a共x1 , l兲 , a共x2 , l兲 , . . . , a共xK , l兲兴 and the signal wave forms of the sources are given by Yardibi et al.: Sparsity constrained deconvolution approaches
si共l兲 关si,1共l兲 , si,2共l兲 , . . . , si,K共l兲兴T, i = 1 , . . . , I. The elements of ei共l兲 and si共l兲 are assumed to be independent zero mean random variables for i = 1 , . . . , I. The steering vector for source k is
1 a共xk, l兲 = C共k,l兲
冤 冥 1 −j x /c e l 1,k 0 x1,k ] 1
x M,k
e
¯a共xk, l兲 = , C共k,l兲
共2兲
冋兺 K
k=1
册
I
1 兺 兩si,k兩2a共xk兲aH共xk兲 ˜a共x⬘兲 I i=1
K
˜ 共xk兲, = 兺 ˜Ak共x⬘兲x
共7兲
k=1
I
ˆ 共 兲 = 1 兺 y 共 兲yH共 兲. R l i l i l I i=1
共3兲
The problem is to estimate the power levels of the sources I ˜xk共xk , l兲 = 共1 / I兲兺i=1 兩si,k共l兲兩2, for l = 1 , . . . , L, k = 1 , . . . , K, and their locations x from the observed data yi共l兲, l = 1 , . . . , L, i = 1 , . . . , I. The delay-and-sum beamformer output power at a given location x⬘ and angular frequency l, l = 1 , . . . , L, is computed as 1 H ˆ 共 兲a ˜a 共x⬘, l兲R l ˜ 共x⬘, l兲, M2
共4兲
I where ˜x共xk兲 共1 / I兲兺i=1 兩si,k兩2 and
˜A 共x⬘兲 = 1 兩a ˜ H共x⬘兲a共xk兲兩2, k M2
共8兲
k = 1, . . . ,K.
Usually, the number of sources, K, is unknown and is replaced by N, which is the number of scanning points in the region. Also, x and x⬘ are scanned over the same set of grid points for simplicity assuming that a certain minimum desired resolution is satisfied. If the power at each scanning point is stacked up to form a power vector ˜y, we arrive at
冤 冥冤 ˜y 共x1兲 ˜y 共x2兲 ]
˜y 共xN兲
˜A 共x 兲 ˜A 共x 兲 . . . ˜A 共x 兲 1 1 2 1 N 1
˜A 共x 兲 ˜A 共x 兲 . . . ˜A 共x 兲 1 2 2 2 N 2 = ] ] ] ˜A 共x 兲 ˜A 共x 兲 . . . ˜A 共x 兲 1 N 2 N N N
冥冤 冥 ˜x共x1兲 ˜x共x2兲 ]
,
冤
冥
x1,ke−jlx1,k/c0 ˜a共xk, l兲 = C共k,l兲 ] . −jlx M,k/c0 x M,ke
共5兲
In aeroacoustic noise measurement applications, the power levels are usually calculated separately for each frequency bin l, l = 1 , . . . , L, and hence in the sequel l will be dropped from the notation for simplicity.
III. REVIEW OF DAMAS
In this section, the inverse equation solved in DAMAS will be obtained from a slightly different perspective than the derivation in the original paper.11 DAMAS uses the delayand-sum beamformer results to obtain the deconvolved source strengths. Hence, substituting Eqs. 共1兲 and 共3兲 in Eq. 共4兲 and dropping the x⬘ term from ˜a, for notational simplicity, we obtain
冋 冉 冋兺
I
冊
册
1 H 1 ˜ A共x兲 兺 sisH AH共x兲 + 2I ˜a 2a M I i=1 i I
1 1 = 2 ˜aH 关a共x1兲si,1 + ¯ + a共xK兲si,K兴关a共x1兲si,1 M I i=1
册
+ ¯ + a共xK兲si,K兴H + 2I ˜a . J. Acoust. Soc. Am., Vol. 123, No. 5, May 2008
共6兲
共9兲
˜x共xN兲
˜ ˜x, each term of which can be expressed as or ˜y = A ˜y n = ˜An,1˜x1 + ˜An,2˜x2 + ¯ + ˜An,N˜xN ,
where,
˜y =
1 ˜y 共x⬘兲 = 2 ˜aH共x⬘兲 M
−jlx M,k/c0
where xm,k is the distance between the mth microphone and ¯ 共xk , l兲储2, i.e., C共k , l兲 is a conthe kth source and C共k , l兲 = 储a stant used to normalize the columns of A共xk , l兲. Note that this does not restrict the following analysis in any way. The sample covariance matrix, or the cross spectral matrix, for a given l, l = 1 , . . . , L, is calculated as
˜y 共x⬘, l兲 =
The cross terms in Eq. 共6兲 are assumed to be negligible which is a resonable assumption when I Ⰷ 1. In DAMAS, this expression is approximated by also neglecting the noise term to obtain
共10兲
where n = 1 , . . . , N. The solution for ˜x can be obtained by using the Gauss–Seidel method and noting that ˜Ann = 1, from ˜ , as follows: the definition of A
冉 冋
˜x共k兲 ˜n − n = max 0,y
n−1
兺 ˜Anj˜x共k兲j + j=1
N
˜A ˜x共k−1兲 兺 nj j j=n+1
册冊
,
共11兲
where k is the current iteration number limited above by a user defined maximum number of iterations, ˜x共0兲 = 0 and the positivity of every element of ˜x is enforced since ˜x represents power. The major drawbacks of this algorithm may be summarized as follows. First of all, neglecting the noise term in Eq. 共7兲 introduces a modeling error since, in almost all practical cases, some kind of noise will contaminate the data. Second, the algorithm requires many iterations, on the order of thousands, to give reasonable results, which can become very time consuming depending on the resolution, i.e., the length of ˜x. Also, the delay-and-sum beamformer has to be implemented before using DAMAS which also means more processing time and effort. Finally, the Gauss–Seidel method is ˜ is diagonally dominant, not guaranteed to converge unless A i.e., for each row, the absolute value of the diagonal term is greater than the sum of absolute values of other terms which is usually not true.12 Despite these drawbacks, the algorithm has been successfully employed in many practical applications. Yardibi et al.: Sparsity constrained deconvolution approaches
2633
Diagonal removal (DR). The first of the abovementioned drawbacks can be overcome by the use of diagonal removal which eliminates the noise occurring in the diˆ by removing the diagonal elements, i.e., making agonal of R them zero.2,10,11 In this case, the delay-and-sum output is calculated as ˜y DR共x⬘兲 =
1 ˜ ˜a共x⬘兲, ˜aH共x⬘兲R M2 − M
共12兲
˜ is obtained by removing the diagonal elements of where R ˜ DR˜x, where ˆ . Then, ˜yDR = A R ˜ADR共x⬘兲 = k
1 ˜aH共x⬘兲关a共xk兲aH共xk兲兴diag.=0˜aH共x⬘兲, M −M 2
共13兲 for k = 1 , . . . , K. In the rest of the paper, DAMAS is applied with diagonal removal in all cases. Otherwise, the performance becomes worse as 2 increases. Consequently, we will denote ˜ DR as ˜y and A ˜ , respectively, in order to simplify ˜yDR and A the notation.
method for choosing the parameter automatically. Also, it is empirically observed that the method in Eq. 共15兲 is not very sensitive to the selection of . In fact, if we let → ⬁, the formulation will reduce to a non-negative least-squares problem. However, using prior knowledge about the sparsity of ˜x will improve the estimate in most cases. The problem in Eq. 共15兲 is a quadratic convex optimization problem which can be solved efficiently via readily available interior point methods31,32 to find the globally optimal solution for ˜x. SelfDual Minimization31 is an extensively used software package in the signal processing community for solving optimization problems over symmetric cones including linear, quadratic, second-order conic and semidefinite programs. The detailed discussion of advanced optimization methods is beyond the scope of this paper and the interested reader is referred to a text on convex optimization.33 B. Estimating the user parameter
In the formulation Eq. 共15兲, constrains the ᐉ1 norm of ˜x, which represents the total power of the signals. Since this value is unknown, a way of determining has to be found for practicality. Let H 2 R = E兵yiyH i 其 = ADA + I,
IV. SPARSITY CONSTRAINED DAMAS „SC-DAMAS… A. Sparsity constrained formulation
Assume that a measured vector ˜y exists which is known ˜ ˜x where A ˜ is known and ˜x to satisfy the linear relation ˜y = A is an unknown quantity that is to be estimated, see Eq. 共9兲. In its simplest form, sparse modeling can be stated as follows: ˜ ˜x . ˜ 储0 subject to ˜y = A minimize 储x
共14兲
Usually, this problem is a combinatorial problem which becomes intractable quickly as the dimension of ˜x increases.17 If the solution is sufficiently sparse, the ᐉ0 norm can be replaced with the ᐉ1 norm to make the problem convex.17 The ˜ is usually ill important point in Eq. 共14兲 is that the matrix A conditioned and not invertible. Otherwise, the solution could ˜ been a square mabe obtained by taking the inverse, had A trix, or by the least-squares method otherwise. Following the above-presented discussion, the problem in Eq. 共9兲 is directly applicable in the sparse signal representation context by observing that ˜x is sparse due to replacing K by N in Eq. 共9兲 and noting that N Ⰷ K. In other words, ˜ contains all the scanning points and only a small since A portion of the region is expected to possess strong sources, ˜x will contain few nonzero elements with respect to its size. Thus, we can immediately think of applying a slightly modified version of LASSO18 to this problem, minimize
˜ ˜x储2 , ˜−A 储y 2
subject to
˜ 储1 艋 , 储x
˜xn 艌 0,
n = 1, . . . ,N,
共15兲
where the modification is to enforce every element of ˜x to be positive. Here, the user parameter is unknown and has to be empirically tuned or estimated from the observation vec˜ . The following section describes a simple tor ˜y and A 2634
J. Acoust. Soc. Am., Vol. 123, No. 5, May 2008
共16兲
where D = E兵sisH i 其, which is a diagonal matrix for uncorrelated sources, is given by
D=
冤
d1 0 ¯
0
0 d2 ¯ ] ]
0 ]
0 ¯ dK
0
冥
,
共17兲
where dk, k = 1 , . . . , K is the power of the kth source. The eigendecomposition of ADAH can be written as ADAH = U⌳UH, where the columns of the unitary matrix U denote the eigenvectors of ADAH and the diagonal elements of the diagonal matrix ⌳ are the corresponding eigenvalues, denoted as 1 艌 2 艌 ¯ 艌 K 艌 0 = ¯ = 0. ⌳ is an M ⫻ M matrix where M Ⰷ K. Then, R can be written as R = U共⌳ + 2I兲UH U⌫UH ,
共18兲
where the diagonal elements of the diagonal matrix ⌫ are ␥1 艌 ␥2 艌 ¯ 艌 ␥K 艌 2 = ¯ = 2. Note that Tr共⌳兲 = Tr共U⌳UH兲 = Tr共ADAH兲 = Tr共DAHA兲 = Tr共D兲 K
= 兺 dk ,
共19兲
k=1
which is the total power of the sources. Here, we have used the fact that the columns in A have unit norm and D is diagonal. In practice, we only have the estimated covariance matrix given in Eq. 共3兲 instead of the true covariance matrix R. ˆ with the ˆ =U ˆ ⌫ˆ U ˆ H denote the eigendecomposition of R Let R ˆ , i.e., the diagonal elements of ⌫ˆ , arranged eigenvalues of R in nonincreasing order. Then, we can determine the user parameter in Eq. 共15兲 as Yardibi et al.: Sparsity constrained deconvolution approaches
ˆ − ␥ˆ I兲, = Tr共⌫ M
共20兲
minimize兵dn其N
n=1
where ␥ˆ M is the smallest diagonal element of ⌫ˆ . We refer to the approach of solving Eq. 共15兲 using the determined in Eq. 共20兲 as the sparsity constrained DAMAS 共SC-DAMAS兲. We will show using numerical examples that SC-DAMAS is computationally more efficient than DAMAS. Note that, SCDAMAS also uses the diagonal removal approach described for DAMAS.
冤 冥冤 ]
=
˜y 共xN 兲 0
˜A 共x 兲 2 1
...
˜A 共x 兲 N 1
˜A 共x 兲 1 2 ]
˜A 共x 兲 2 2 ]
...
˜A 共x 兲 N 2 ]
˜A 共x 兲 ˜A 共x 兲 . . . ˜A 共x 兲 1 N0 2 N0 N N0
冥冤 冥 ˜x共x1兲 ˜x共x2兲 ]
,
VI. CORRELATED SOURCES
˜x共xN兲
DAMAS, SC-DAMAS, and CMF algorithms all assume that the noise sources are uncorrelated so that the cross terms in Eq. 共6兲 can be neglected. However, if the sources are correlated, this assumption no longer holds and the algorithms have to be modified to deal with this case. After briefly reviewing DAMAS-C15 for completeness, we will introduce the extension of the CMF approach to the correlated case.
共21兲
A. DAMAS-C
DAMAS-C extends DAMAS to deal with correlated signals.15 Similar to DAMAS, this algorithm uses Eq. 共6兲 but without ignoring the cross terms. In this case, we end up with N共N + 1兲 / 2 unknowns and thus have to evaluate the delayand-sum beamformer at more points. For this purpose, the cross-beamformer output is15 y´ 共x⬘,x⬙兲 =
y´ 共x1,x2兲
A´1,1共x1,x1兲 . . . A´1,N共x1,x1兲 . . . A´N,N共x1,x1兲
]
A´1,1共x1,x2兲 . . . A´1,N共x1,x2兲 . . . A´N,N共x1,x2兲
y´ 共xN,xN兲
or
]
]
]
]
]
]
A´1,1共xN,xN兲 . . . A´1,N共xN,xN兲 . . . A´N,N共xN,xN兲
J. Acoust. Soc. Am., Vol. 123, No. 5, May 2008
1 H ˆ ˜a共x⬙兲, ˜a 共x⬘兲R M2
共23兲
which is similar to Eq. 共4兲. Then, the inverse problem can be defined as in the following:
y´ 共x1,x1兲
]
共22兲
which is a quadratic convex optimization problem where is defined in Eq. 共20兲, dn is the noise power level for grid n, N n = 1 , . . . , N, and D is an N ⫻ N diagonal matrix with 兵dn其n=1 as its diagonals, see Eq. 共17兲. Also recall that A = 关a共x1兲 , a共x2兲 , . . . , a共xN兲兴. The idea behind this method is quite intuitive in the sense that it basically tries to fit the unknown signal powers and the noise power to the model in Eq. 共16兲 such that the solution is sparse. In contrast to deletˆ , this method tries to extract the noise ing the diagonals of R and use the signal components in the diagonal. Moreover, this formulation does not require the implementation of the delay-and-sum beamformer as an initial step and it converges quickly thanks to the convex formulation. Similar to SCDAMAS, this algorithm also turns out to be quite insensitive to .
Instead of using the sample covariance matrix to obtain delay-and-sum estimates and then trying to deconvolve the results, we introduce a method for estimating source locations and strengths based directly on the sample covariance matrix in Eq. 共3兲 and the steering vector in Eq. 共2兲. Specifically, the covariance matrix fitting approach determines 20 N and 兵dn其n=1 via
冤 冥冤
兺 dn 艋 ,
2 艌 0,
V. COVARIANCE MATRIX FITTING
y´ 共x2,x1兲
n = 1, . . . ,N,
n=1
˜ has become a fat matrix, i.e., the where N0 ⬍ N. Note that A number of its rows is smaller than the number of its columns. It will be shown in Sec. VII that this approach can save a significant amount of computation time without degrading the performance. A similar approach cannot be used with DAMAS since DAMAS requires N equations to solve for the N unknowns in ˜x. Formulating the same problem as a sparse problem alleviates the need for N equations.
y´ 共x1,xN兲 =
dn 艌 0,
subject to
The delay-and-sum estimate ˜y usually contains redundant information due to its wide beamwidth. Therefore, it is not always necessary to beamform at all N locations in the region when using SC-DAMAS. Beamforming at fewer points reduces the size of ˜y and increases the speed of SCDAMAS. In this case, Eq. 共9兲 becomes ˜A 共x 兲 1 1
ˆ − ADAH − 2I储2 , 储R F N
C. A more efficient implementation of SC-DAMAS
˜y 共x1兲 ˜y 共x2兲
,2
冥冤
x´共x1,x1兲 x´共x1,x2兲 ]
冥
x´共x1,xN兲 , x´共x2,x1兲 ] x´共xN,xN兲
Yardibi et al.: Sparsity constrained deconvolution approaches
共24兲
2635
FIG. 1. The beamforming maps of the actual sources, DAS, DAMAS, SC-DAMAS, and CMF with three different settings as follows: I = 10 000, 共a兲 f = 10 kHz and 2 = 0, f = 15 kHz: 共b兲 2 = 0, 共c兲 2 = 100. The 2D plots represent the beamforming area and the power levels are in decibels.
´ x´ , y´ = A
共25兲
˜ H共x⬘兲a共xk兲aH共xl兲a ˜ 共x⬙兲, k,l where Ák,l共x⬘ , x⬙兲 = 共1 / M 2兲a = 1 , . . . , K. Although the derivation is straightforward, this 2636
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´ , and x´ equation is cumbersome. Note that the sizes of y´ , A are squares of the sizes of their uncorrelated counterparts ˜y, ˜ , and ˜x. This equation is solved for x´ with a very similar A Yardibi et al.: Sparsity constrained deconvolution approaches
FIG. 2. The beamforming maps of the actual sources, DAS, DAMAS, SC-DAMAS, and CMF with two different settings as follows: f = 20 kHz, I = 500 and 共a兲 2 = 0, 共b兲 2 = 5. The 2D plots represent the beamforming area and the power levels are in decibels.
algorithm to DAMAS. Again, positivity is enforced as in Eq. 共11兲. Yet, now there may be complex terms arising from the cross products and a suitable constraint has to be found for these terms too. The correlations are assumed to be in phase and real in DAMAS-C.15 With this assumption, the positivity is enforced on every element of x´ , see Eq. 共11兲, as the iterations proceed. As a final note, diagonal removal is also applied with DAMAS-C in our simulations.
minimizeD,2 subject to
ˆ − ADAH − 2I储2 , 储R F
D Ɒ 0,
tr共D兲 艋 ,
2 艌 0,
共26兲
The extension of CMF to the correlated case is as follows:
which looks simpler than the DAMAS-C formulation. Note that D is no longer a diagonal matrix and when the sources are uncorrelated, Eq. 共26兲 reduces to the CMF formulation, Eq. 共22兲. CMF-C is a semidefinite quadratic program and can also work with complex correlations between source signals in contrast to DAMAS-C. is determined as in Eq. 共20兲. Both CMF-C and DAMAS-C require extreme computation times as the number of grid points increases since the
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B. CMF-C
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FIG. 3. The beamforming maps of the actual sources, DAS, DAMAS, SC-DAMAS, and CMF with two different settings as follows: f = 5 kHz, I = 500 and 共a兲 2 = 0, 共b兲 2 = 4. The 2D plots represent the beamforming area and the power levels are in decibels.
number of unknowns in both is approximately N2. Nevertheless, a parallel implementation can make these algorithms usable in practice.15 We will evaluate these algorithms only for a small grid size in Sec. VII. Note that a similar approach to SC-DAMAS can be used to solve the DAMAS-C problem given in Eq. 共25兲 but this issue is not further explored here since our experience is that even in the uncorrelated case CMF usually results in better performance than SCDAMAS. In results not shown here, it was observed that this is true in the correlated case, too. 2638
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VII. NUMERICAL EXAMPLES
In this section, we evaluate the performances of DAMAS, SC-DAMAS, and CMF for the SADA array.1,34 SADA is a two-dimensional 共2D兲 array consisting of 33 microphones and has a maximum diagonal aperture size of 19.71 cm. A microphone is located at the center and the others are arranged in four circles of eight microphones each where the diameter of each circle is twice the diameter of the circle it encloses. We assume that the SADA is at a distance Yardibi et al.: Sparsity constrained deconvolution approaches
TABLE I. Characteristics of the acoustic imaging algorithms.
Resolution Sensitivity to noise Computation Time Max. number of variables
TABLE III. Speeding up of SC-DAMAS for the example of Fig. 1共b兲. Timing values are in seconds.
DAS
DAMAS
SC-DAMAS
CMF
Low Low Low High
Medium Medium High Medium
High Medium Medium Low
High Low Medium Low
of 1.52 m from the region of interest similar to the DAMAS paper.10,11 The span for the x axes and y axes is from −25.4 to 25.4 cm and the resolution in both directions is 2.54 cm. We simulate the received signal at each microphone according to Eq. 共1兲. The noise components ei共l兲 in Eq. 共1兲 are assumed to be uncorrelated with the source signals and distributed as circularly symmetric i.i.d. complex Gaussian random variables with mean zero and variance 2. The signal wave forms are also distributed as circularly symmetric i.i.d. complex Gaussian random variables with mean zero and a certain power level which is assumed to be 25 dB. This value is chosen without loss of generality since 2 will be varied throughout the experiments. There are three parameters which affect the performances of the algorithms directly: number of snapshots I, noise variance 2, and frequency of interest f = / 共2兲. The incoherence assumption breaks down as the number of snapshots decreases since the cross terms in Eq. 共6兲 are no longer negligible and also the additional noise term in Eq. 共6兲 is no longer 2I. Noise affects all the algorithms negatively as in all applications. Finally, the frequency determines the resolution of the algorithms since as the frequency increases, the difference in the steering vectors for nearby sources increases and hence it is easier for the algorithms to discriminate them. We will first evaluate DAMAS, SC-DAMAS, and CMF for uncorrelated sources and then evaluate DAMAS-C and CMF-C for correlated sources. We start our simulations with a relatively easy scenario where the number of snapshots is large, I = 10 000, and 2 = 0 in which case the assumptions of the algorithms are almost correct. The resulting beamforming maps are shown in Figs. 1共a兲 and 1共b兲, where the horizontal axis and the vertical axis represent the 2D scanning plane and the power levels are represented in a gray color scale over a span of 10 dB. As expected, the delay-and-sum algorithm merges the peaks as if there were a single source. CMF works quite well and SC-DAMAS and DAMAS show good performance when the frequency is high. Next, the noise variance 2 is increased to 100. As shown in Fig. 1共c兲, CMF is more robust to noise than DAMAS and SC-DAMAS. Note that for all the examples in this section, DAMAS has been run for 10 000 iterations after which no significant improvement was observed.
N0 Time
441 24.0
196 12.7
121 4.2
81 2.8
36 1.3
Next, we decrease the number of snapshots to 500 and increase the frequency to 20 kHz to resolve a more complicated source distribution. The results are shown in Fig. 2 for 2 = 0 and 5. The location estimates of SC-DAMAS for some of the sources are inaccurate and DAMAS is not able to discriminate the sources. CMF, however, is able to recover the sources reasonably well in both cases. If the frequency is decreased to f = 5 kHz, DAMAS performance degrades significantly as also observed in other studies.16 Figure 3 shows the imaging maps for 2 = 0 and 4 with I = 500. DAMAS is not able to recover the source location for f = 5 kHz even after 50 000 iterations. We again observe offsets in the location estimates of SC-DAMAS and when the noise is increased, the results become worse. This is due to the fact that SC-DAMAS is solving the same DAMAS inverse problem. In Fig. 3共a兲, the highest outlier for CMF is 5 dB below the actual signal levels and in Fig. 3共b兲, CMF provides the best performance. This performance gain can be attributed to the different formulation of CMF than DAMAS and SC-DAMAS. Recall that CMF does not rely on the delayand-sum beamformer estimates and does not delete the diagonals. Note that the source distribtution considered for the last case is simpler than the other two since the frequency is low. A qualitative assessment of the algorithms is given in Table I. The computational complexity of DAMAS was higher than that of SC-DAMAS and CMF in our simulations. Note that the public domain solver31 we use for finding the optimal solutions to the SC-DAMAS and CMF problems works up to N = 1000 grid points when the signals are complex valued. However, a commercial software designed for this purpose can go up to many more variables and hence the formulations are applicable to higher resolutions if desired. Table II shows the computation times of each algorithm for the examples considered in this section. We observe that SCDAMAS is the fastest and CMF takes almost thrice the time of SC-DAMAS. DAMAS takes almost twice the time of CMF and eight times the time of SC-DAMAS. Furthermore, as mentioned in Sec. IV, the sparse formulation of SCDAMAS allows us to reduce the size of ˜y in Eq. 共9兲 which provides more improvements in speed. In Table III, we noted the computation time of SC-DAMAS for different values of N0 for the example considered in Fig. 1共b兲. The performance of SC-DAMAS undergoes only a minor degradation when
TABLE II. Computation times for DAMAS, SC-DAMAS, and CMF in seconds on personal computer, 2.0 GHz dual core processor, 2 Gbytes of RAM.
DAMAS SC-DAMAS CMF
Fig. 1共a兲
Fig. 1共b兲
Fig. 1共c兲
Fig. 2共a兲
Fig. 2共b兲
Fig. 3共a兲
Fig. 3共b兲
160 24 66
165 24 65
161 24 71
160 23 69
160 24 65
180 21 75
162 22 73
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DAMAS−C
DAMAS−C
DAMAS−C
20
20
20
15
15
10
15
10
5
0
Power (dB)
25
Power (dB)
25
Power (dB)
25
10
5
5
10
15
20
25
30
0
35
5
5
10
15
n
20
25
30
0
35
CMF−C
15
10
20
25
30
35
0
35
10
5
15
30
15
10
5
25
Power (dB)
20
Power (dB)
20
Power (dB)
20
15
20
CMF−C 25
10
15
CMF−C 25
5
10
n
25
0
5
n
5
5
10
15
20
25
30
35
0
5
10
15
20
n
n
n
(a)
(b)
(c)
25
30
35
FIG. 4. The power estimates with DAMAS-C and CMF-C with three different settings as follows: I = 500, 共a兲 f = 10 kHz and 2 = 0, 共b兲 f = 10 kHz and 2 = 2, 共c兲 f = 20 kHz and 2 = 0. DAMAS-C was run for 10 000 iterations. The y axis represents the power values in decibels and the x axis represents the index corresponding to the 2D grid point. The true source locations and powers are marked with circles.
N0 艌 36. Below that, the algorithm does not provide accurate results. The savings are huge and the problem can be solved in almost 1 s when N0 = 36, which should be compared with 167 s for DAMAS and 62 s for CMF keeping in mind that CMF provides better results than SC-DAMAS and SCDAMAS provides better results than DAMAS. In any case, SC-DAMAS offers a fast way of solving Eq. 共9兲. Finally, the performances of DAMAS-C and CMF-C are evaluated when there are two coherent acoustic signals in the scanning region. We assume a simple scenario which consists of 36 grid points between −12.7 and 12.7 cm with 5.08 cm increments in both x and y directions. Thus, the number of unknowns is 666 for each algorithm. The results are shown in Fig. 4. In these plots, rather than showing the 2D images, we plot the one-dimensional versions where the horizontal axis corresponds to the grid number n, where n = 1 , . . . , N, and the vertical axis is the power levels in decibels since this way the power estimates are seen more clearly for this case. The actual noise locations and strengths are indicated with circles in the plots. We observe that DAMAS-C does not provide accurate estimates for the lower frequency scenario. But when the frequency increases, it can discriminate the sources. Note that results not shown here showed that DAMAS and CMF, which were designed for uncorrelated sources, cannot separate the two coherent sources, as expected. CMF-C can correctly identify the two sources even when 2 ⫽ 0 as shown in Fig. 4共b兲. The outlier occurring in this case is almost 16 dB below the correct es2640
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timates and is reasonable when compared to DAMAS performance. DAMAS-C and CMF-C take much more time than DAMAS and CMF due to the larger number of unknowns in the coherent case. As mentioned before, it is also possible to extend SC-DAMAS to the correlated case to achieve computational savings by sacrificing some accuracy as compared to CMF-C. VIII. CONCLUSIONS
In this paper, a sparsity constrained deconvolution algorithm 共SC-DAMAS兲 and a sparsity constrained covariance matrix fitting algorithm 共CMF兲 were presented for the recovery of acoustic source locations and strengths. The former algorithm is an extension of DAMAS and tries to solve the same basic equation by exploiting sparsity. Similar to DAMAS, SC-DAMAS employs diagonal removal to mitigate the effects of noise, whereas the CMF algorithm eliminates noise without deleting the diagonals of the covariance matrix completely. Also, DAMAS and SC-DAMAS algorithms require the implementation of the delay-and-sum method and depend on the performance of this method. On the other hand, CMF is independent of the delay-and-sum algorithm. DAMAS, SC-DAMAS, and CMF were evaluated using synthetic data and the CMF approach showed the best performance both in terms of accuracy and robustness to noise. An alternative implementation of SC-DAMAS was provided which offers a very fast algorithm as compared to Yardibi et al.: Sparsity constrained deconvolution approaches
xm,k ⫽ distance between the mth microphone and the kth source N ˜x ⫽ vector of 兵x ˜ 共xn兲其n=1 ˜xi ⫽ ith element of ˜x ˜x共x⬘兲 ⫽ the actual source power level at x⬘ yi ⫽ measurement vector at snapshot i ⫽ user parameter for SC-DAMAS and CMF 2 ⫽ noise variance ⫽ angular frequency 储 · 储0 ⫽ ᐉ0 norm: the number of nonzero elements 储 · 储1 ⫽ ᐉ1 norm: the absolute sum of the elements 储 · 储2 ⫽ ᐉ2 norm 储 · 储F ⫽ Frobenius norm 共·兲T ⫽ transpose of the argument 共·兲H ⫽ conjugate transpose of the argument 共·兲 Ɒ 0 ⫽ the matrix argument is positive semi-definite 关·兴diag.=0 ⫽ diagonal of the matrix argument is zero
DAMAS and CMF. Keeping in mind the better performance of SC-DAMAS than DAMAS, this could be a very practical deconvolution algorithm for aeroacoustic applications. DAMAS is observed to provide relatively worse performance as the frequency decreases. The number of snapshots affects all the algorithms significantly. Usually, large amounts of data are collected in aeroacoustic applications and the values used in the simulations are not far from being practical. An extension of CMF, CMF-C, was also proposed to deal with correlated sources and it was shown to provide better performance than DAMAS-C. The next step is to evaluate the algorithms on real data. ACKNOWLEDGMENTS
This work was supported in part by NASA under Grant No. NNX07AO15A and the Swedish Science Council 共VR兲. LIST OF SYMBOLS a共x⬘兲 A共x⬘兲 ˜ A ˜A i,j BP CMF CMF-C c0
⫽ steering vector for location x⬘ ⫽ steering matrix ⫽ matrix used in the inverse problem ˜ ⫽ 共i , j兲th element of A
⫽ ⫽ ⫽ ⫽
DAS ⫽ DAMAS ⫽ DAMAS-C DR E兵.其 e f FFT FOCUSS i.i.d. I K L LASSO
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
LORE M N R ˆ R
⫽ ⫽ ⫽ ⫽ ⫽
SADA ⫽ SC-DAMAS ⫽ SVD ⫽ sk ⫽ Tr共·兲 ⫽ xk ⫽ x ⫽ x⬘ , x⬙ ⫽
Basis pursuit Covariance matrix fitting extension of CMF to the correlated case sound propagation velocity in air, taken as 343 m / s delay-and-sum beamformer deconvolution approach for the mapping of acoustic sources extension of DAMAS to the correlated case diagonal removal expectation operation noise focusing frequency fast Fourier transform focal underdetermined system solution independent identically distributed number of snapshots number of sources number of frequency bins least absolute shrinkage and selection operator localization and optimization of array results number of microphones in the array number of scanning points in the region theoretical covariance matrix sample covariance matrix 共cross spectral matrix兲 small aperture directional array sparsity constrained DAMAS singular Value Decomposition signal wave form of source k trace of the matrix argument 3D location of source k vector of all source locations arbitrary 3D locations in space
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