Underwater sound receiver with optimized covariance matrix

Abstract

Underwater sound receiver (20) with the following features: a plurality of sound transducers (22), each designed to convert incoming sound waves (24) into a signal (26); a signal processor (28) which is trained, based on the signals (26) an empirical covariance matrix R = Cov (X) = 1 N − 1 ∑ i = 1 N (xi − x ¯) (xi − x ¯)' to determine and to optimize the empirical covariance matrix to obtain an optimized covariance matrix, where the optimized covariance matrix S=αS1 + (1 - α)S2 has a first Estimator S1 = λ1µ1 + (1 - λ1)R with µ̂ = tr(R) / N and λ ^ 1 = ∑ i = 1 p ∑ j = 1, i ≠ jp Var ( R ) ij + ∑ j Var ( R ) jj ∑ i = 1 p ∑ j = 1, i ≠ jp | Rij | 2 + ∑ j | R jj − μ ^ | 2 as well as a second estimator S2 = λ2dR + (1 - λ2)R with λ ^ 2 = ∑ i = 1 p ∑ j = 1, i ≠ jp Var ( R ) ij ∑ i = 1 p ∑ j = 1, i ≠ jp | Rij | 2 exhibits where a ^ = ∑ i = 1 p ∑ j = 1 p ( Var ( S 2 ) − Cov ( S 1 , S 2 ) − Bias ( S 2 ) E { S 1 − S 2} ) if E { ‖ S 1 − S 2 ‖ F 2} is selected, whereby the signal processor is configured for the counter Z = ( 1 + λ ^ 1 − λ ^ 2 ) λ ^ 2 Var ( dR ) + ( 1 − λ ^ 2 ) ( λ ^ 1 − λ ^ 2 ) Var ( R ) + λ ^ 2 ( λ ^ 1 − λ ^ 2 ) ( E { R} − E { dR} ) ⊙ ( E { R} − E {dR}). to evaluate wherein the signal processor (28) is configured to perform a direction determination of the sound waves (24) based on the optimized covariance matrix.

Description

[0001] The invention relates to the directional formation in underwater sound receivers.

[0002] Various methods, such as adaptive beamforming (ABF), use a covariance matrix to obtain information about the second moments of the processed data. Beamforming methods include robust capon or robust MPDR (Minimum Power Distortionless Response), which use the empirical covariance matrix. However, when using the empirical covariance matrix, these methods have the disadvantage that a nearby active sonar that regularly emits sound waves, so-called pings, into the environment "blinds" an underwater sound receiver, i.e. other targets disappear. This means that due to the large amount of energy emitted by the ping, information from incoming sound waves cannot be detected, even at times that do not coincide with the ping. This effect is further amplified by the non-stationary pulse.

[0003] US 2015 / 0 226 831 A1 discloses an apparatus and a method for sound processing with the aim of obtaining a convergent estimation error.

[0004] US 2012 / 0 093 344 A1 discloses a beamformer for sensor arrays with a method for finding a global optimal beam pattern for a spherical array using multiple boundary conditions.

[0005] US 2009 / 0 257 312 A1 discloses an improved sonar system that can operate autonomously.

[0006] US 2006 / 0 133 211 A1 discloses a method for acoustic target tracking using a horizontal line array.

[0007] US 9 559 417 B1 discloses adaptive beamforming using a covariance matrix estimated on the basis of sensor data.

[0008] US 6 424 596 B1 discloses a method and system for significantly reducing acoustic noise from near-surface sources using an array processing technique that utilizes Multiple Signal Classification (MUSIC) beamforming and the Lloyd's Mirror interference pattern at very low frequencies.

[0009] The object of the present invention is therefore to create an improved concept for directional sound generation in underwater sound receivers.

[0010] This problem is solved by the subject matter of the independent patent claims. Further advantageous embodiments are the subject matter of the dependent patent claims.

[0011] Embodiments show an underwater sound receiver with a plurality of sound transducers, each configured to convert incoming sound waves into a signal. A signal processor is configured to determine an empirical covariance matrix based on the signals and to optimize the empirical covariance matrix to obtain an optimized covariance matrix.

[0012] The optimized covariance matrix S=αS1 + (1-α)S2 has a first estimator S1 = λ µ1 + (1- λ1)R with µ̂ = tr(R) / N and λ^1=∑i=1p∑j=1,i≠jpVar(R)ij+∑jVar(R)jj∑i=1p∑j=1,i≠jp|Rij|2+∑j|Rjj−μ^|2. and a second estimator S2 = λ2dR+(1-λ2) R with λ^2=∑i=1p∑j=1,i≠jpVar(R)ij∑i=1p∑j=1,i≠jp|Rij|2 on, whereby a^=∑i=1p∑j=1p(Var(S2−Cov(S1,S2)−Bias(S2)E{S1−S2}))E{‖S1−S2‖F2} The signal processor is designed to be used for the counter Z=(1+λ^1−λ^2)λ^2Var(dR)+(1−λ^2)(λ^1−λ^2)Var(R) +λ^2(λ^2−λ^1)(E{R}−E{dR})⊙(E{R}−E{dR}) to evaluate and for the denominator E{‖S^1−S^2‖F2}=E{‖λ^1(μ^1−dR)‖F2+‖(λ^2−λ^1)(R−dR)‖F2} to evaluate.

[0013] Furthermore, the signal processor is designed to determine the direction of the sound waves based on the optimized covariance matrix. These signals are also referred to as output signals or stave signals.

[0014] The idea is to improve the disadvantages of adaptive beamforming using an empirical covariance matrix by optimizing the covariance matrix. Various approaches are presented, each of which improves beamforming on its own. A combination of two or more of these approaches results in a further improvement in beamforming.

[0015] For example, very loud targets can lead to an adjustment of the antenna's gain in the (non-optimized) empirical covariance matrix. In this case, the gain may not be adjusted quickly enough. This can lead to clipping (overmodulation) of the signals. In this case, the covariance matrix can no longer be reliably estimated. However, it can also happen that adjusting the gain for the loud signals causes an attenuation of the very quiet signals. In this case, the weak targets would potentially become even weaker. Optimization mitigates these disadvantages.

[0016] Embodiments show that the signal processor performs a time-frequency transformation of the signals in order to obtain a spectral mapping of the signals and to determine the empirical covariance matrix based on the spectral mapping of the signals. Furthermore, the signal processor is designed to use a window, in particular a Von Hann window, for the time-frequency transformation, which uses a smaller factor for multiplying by the signal for the first samples, in particular the first 5 samples, than a Hamming window. When converting the signals into spectra in conventional ABF, a Hamming window is typically used. For rapidly changing signals, e.g., when the pulse starts abruptly, a window that begins with very small values ​​can sometimes deliver better results than the Hamming window used.The adjusted window can be used to reduce artifacts caused by non-stationary signals in particular.

[0017] Further embodiments show that the signal processor is configured to form the empirical covariance matrix based on sample values ​​of the signal or a spectral map of the signal, wherein the signal processor is configured to normalize the sample values. Both non-stationarity and levels can lead to artifacts in the sonar images. The primary challenge is adjusting the level, as this increases the background to such an extent that other targets disappear. To reduce this influence, a new normalization of the input data or spectra is used, whereby stationary signals are emphasized more strongly than loud, short signals. This normalization is optionally combined with an adjustment of the window function. The normalization reduces artifacts caused by non-stationary signals.

[0018] In embodiments, the signal processor is configured to optimize the empirical covariance matrix using a weighted combination of a first optimization method and a second optimization method, wherein the weighted combination is weighted using a first weighting factor. The optimization can be performed using regularization. The signal processor can perform the optimization using the first optimization method (S1) by diagonal loading of the empirical covariance matrix. Furthermore, the signal processor can perform the optimization using the second optimization method (S2) by diagonal shrinkage of the empirical covariance matrix.

[0019] To optimize the empirical covariance matrix, the signal processor can overlay the empirical covariance matrix with the weighted combination of the first optimization method and the second optimization method. The signal processor can perform the overlay using a second weighting factor, wherein the second weighting factor is limited to a range between 0 and 1, in particular between 0 and 0.8, preferably between 0 and 0.7. By optimizing the empirical covariance matrix using the first and / or the second optimization method, the background in the sound waves is increased. The second weighting factor is used to limit the background so that it is not increased too much, so that the target (useful signal) is not overpowered by the background.

[0020] Analogously, a method for operating an underwater sound receiver is shown with the following steps: using a plurality of sound transducers, each configured to convert incoming sound waves into a signal; determining an empirical covariance matrix based on the signals; optimizing the empirical covariance matrix to obtain an optimized covariance matrix; performing a direction formation of the signals based on the optimized covariance matrix.

[0021] Preferred embodiments of the present invention are explained below with reference to the accompanying drawings. In the drawings: Fig. 1: a schematic representation of an underwater sound receiver.

[0022] Before exemplary embodiments of the present invention are explained in more detail below with reference to the drawings, it is pointed out that identical, functionally equivalent or equivalent elements, objects and / or structures in the different figures are provided with the same reference numerals, so that the description of these elements shown in different exemplary embodiments is interchangeable or can be applied to one another.

[0023] Fig.Figure 1 shows a schematic representation of an underwater sound receiver 20. Underwater sound receiver 20 comprises a plurality of sound transducers 22. The sound transducers 22 receive sound waves 24 from their surroundings and convert them into (electrical) (output) signals. A first sound transducer 22a thus generates a first signal 26a. A second sound transducer 22b generates a second signal 26b. A p-th sound transducer 22c generates a p-th signal 26c. The signals are represented as a function of their amplitude over time—i.e., as a time signal. Alternatively, the signals are transformed into the frequency domain, in which case the frequency amplitude of a frequency is represented over time.

[0024] The signals are sampled at predetermined times. For a time period with N sampling times, the sample values ​​for the first signal 26a are x11,x21, to xN1. For the second signal 26b, the sample values x11,x22, to xN2. For the p-th signal 26c, the sample values x1p, x2p, to xNp. The samples at a sampling time i are stored as a vector x i This means that the samples at the first sampling time are summarized in the vector x1 and the samples at the N-th sampling time are summarized in the vector x N summarized.

[0025] The sample values ​​x iare processed by a signal processor 28 to generate a representation 30, in particular a visual representation, of the environment of the underwater sound receiver 20 from the sound waves 24 or the signals. It should be noted that the sampling of the signals from the sound transducers and any time-frequency transformation of the signals can already be carried out by means of the signal processor 28. One processing step carried out in the signal processor 28 is beamforming. By means of beamforming, the directions from which a sound is coming can be determined, so that the direction in which a sound source is located can be determined. This process is also referred to as bearing taking. The visual representation 30 is typically a visual representation of the received energy from the sound waves over the bearing / direction. An imaging unit, for example a monitor, can output the visual representation 30.

[0026] The following describes the processes that the signal processor can perform.

[0027] Various methods, such as adaptive beamformers, require information about the second moments of the data to be processed, namely the covariance matrix. However, this matrix is ​​usually unknown a priori and must be estimated from the available measurement data. The so-called empirical covariance matrix is ​​often used as an unbiased estimator. However, with small sample sizes and large-scale covariance matrices, the variance of this estimator can be so large that the estimation result is unusable for practical applications. Therefore, the following describes how large covariance matrices can be estimated from only a small amount of measurement data.

[0028] The basis for the calculation or determination of the covariance matrix are N ∈ ℕ independent observations x i , i = 1, ..., N of a p ∈ ℕ dimensional process X ∈ ℝpxN These observations can, for example, be N samples from p=192 staves (acoustic transducers) of the (flank) array (of acoustic transducers). Likewise, N values ​​from one of 128 frequency bands of the corresponding Fourier transform of the signals from the p=192 staves of the flank array can be present. The number of frequency bands and staves is chosen only as an example. In general, the covariance of X is given by Cov(X)=E{(X−E{X})(X−E{X})'} where E{} denotes the expected value and ()' stands for transpose.

[0029] In real-world applications, the covariance matrix Cov(X) is unknown and must be estimated from a finite number of observations. A frequently used estimator is the empirical covariance matrix. However, for a small number of observations, especially when N ≤ p , this has disadvantages compared to other methods. For this reason, in addition to the empirical covariance matrix, other estimators are presented here. These are based on the approach that the empirical covariance is improved through regularization, resulting in an optimized covariance matrix.

[0030] To calculate the empirical covariance matrix, equation 4.1 is R=Cov(X)=1N−1∑i=1N(xi−x¯)(xi−x¯)' solved, where x is the mean x¯=1N∑i=1Nxi designated.

[0031] This estimator R for the covariance matrix has many disadvantages. These include the following: For p > N, R does not have full rank, meaning R cannot be inverted. For the case where p and N are approximately equal, where p ≤ N, R is invertible but may be numerically unstable. The mean square error (MSE) decreases with the number of observations N. For this reason, the MSE is too large for some applications when there are few observations.

[0032] For this reason, the optimization of R, i.e., the empirical covariance matrix, is presented below. "Optimized" means that the MSE of the optimized estimator should be smaller than that of the empirical covariance matrix R. The optimized covariance estimator should also be positive definite, meaning all eigenvalues ​​should be non-zero and invertible.

[0033] The starting point in this case is that the empirical covariance matrix R is overlaid with another estimator S in order to obtain an estimator S optimized with respect to the MSE. n To do this, a linear superposition of the form Sn=λS+(1−λ)R is used, where λ ∈ [0,1]. Accordingly, either both estimators are combined or, in the extreme case, only one of the two estimators is used.

[0034] The crucial point in this superposition of two different estimators is that the estimator S nafter superposition, the MSE should be superior to both initial estimators. To achieve this, the so-called regularization parameter λ should be chosen optimally. This can be done analytically (see, for example, "A well-conditioned estimator for large-dimensional covariance matrices" in Journal of Multivariate Analysis, Vol. 88, pp. 365-411, O. Ledoit and M. Wolf, 2003), which eliminates the need for a computationally intensive numerical search for the optimal value of λ.

[0035] Using the Frobenius norm ‖A‖F2:=∑i=1m∑j=1n|aij|2 for any matrix A ∈ ℂ mxn the optimal value λ 0 of λ by minimizing the MSE. The optimal value of the regularization parameter λ is λ0=∑i=1p∑j=1p(Var(R)−Cov(S,R)−Bias(R)E{S−R})ijE{‖S−R‖F2}

[0036] The operators Var(.), Cov(.,.), Bias(.) and E{.} used for matrices are to be applied element by element.

[0037] Furthermore, equation 4.6 presents the problem that the variance of R, the covariance of S with R, the bias of R and the expected value of S - R are not known, but must be replaced by appropriate estimates: λ^=∑i=1p∑j=1p(Var(R)−Cov(S,R)−Bias(R)E^{S−R})ijE^{‖S−R‖F2}

[0038] Here, ̂ indicates that this is only an estimate and not the true value. The rest of this document presents how the solution can be determined in principle using Equation 4.6. However, this description is purely theoretical, since the estimates of the stochastic moments required for implementation according to Equation 4.7 are explicitly needed. This means that in order to evaluate Equation 4.7, calculation rules for Var(R), Cov(S, R), Bias(R), and the two expected values ​​must be established. This is, however, known to the person skilled in the art.

[0039] Since the evaluation of equation 4.7 may result in λ ̂ > 1 or λ ̂ < 0 is replaced by λ^final=min(max(0,λ^),1) the optimal value of λ is restricted to the interval [0,1].

[0040] In the following two sections, the two approaches for the estimator S of the covariance matrix S=μ1 and S=1⊙R Here, 1 denotes a matrix with ones on the diagonal and zeros on the secondary diagonals. The estimator in equation 4.9 is a diagonal matrix with identical entries, and equation 4.10 is a diagonal matrix with the main diagonal of R. The parameter µ satisfies µ ∈ ℝ + and ⊙ denotes component-wise or point-wise multiplication (A ⊙ B) ij = A ij · B ij for A ∈ ℂ mxn and B ∈ ℂ mxnwith i = 1, ..., and j = 1, ..., n, where n, m ∈ ℕ. This component-wise multiplication is also called the Hadamard product.

[0041] In both cases, the empirical covariance matrix R is used as the starting point. In the first case, the diagonal of R is loaded according to Equation 4.4, which is also called diagonal loading. The second case is called diagonal shrinkage.

[0042] For the use of the estimator S according to equation 4.9, the new estimator S1 according to equation 4.4, which is optimized with respect to the MSE, is given by S1=λ1μ1+(1−λ1)R

[0043] This ensures that S1 is positive definite, since the diagonal always increases. As before, λ1 ∈ [0,1]. Advantageously, λ1 ∈ [0,0.8] or λ1 ∈ [0,0.7] can be chosen. λ1 and subsequently λ2 are also referred to as the second weighting factor.

[0044] The optimal value for µ can be determined independently of λ1. The starting point for determining the optimal value µ̂ of µ is: μ^=minμ‖R−μ1‖F2

[0045] Since the Frobenius norm of the identity matrix 1 is 1 by definition, this term can be transformed to μ=arg minμ∞‖R‖F2−2μ〈R,1〉+μ2

[0046] In this case, (A, B) = tr(AB') / n denotes the scalar product of two matrices A ∈ ℂ mxn and B ∈ ℂ mxn This directly results in the condition for the optimal value of µ: 〈R,1〉=μ^

[0047] This results in μ^=tr(R) / N.

[0048] The estimation of the optimal regularization parameter according to equation 4.7 is given by λ^1=∑i=1p∑j=1,i≠jpVar(R)ij+∑jVar(R)jj∑i=1p∑j=1,i≠jp|Rij|2+∑j|Rjj−μ^|2.

[0049] Thus, equation 4.11 and equation 4.16 can be regarded as automatic diagonal loading, where the optimal regularization parameters µ̂ and λ ̂ 1 according to equation 4.14 and equation 4.16 from R.

[0050] The optimized estimator of the covariance matrix is ​​positive definite, since µ̂ is always greater than zero and the MSE is smaller than the MSE of the empirical covariance matrix R.

[0051] As a second possibility for optimizing the empirical covariance matrix R, according to equation 4.10, the combination S2=λ2dR+(1−λ2)R used. Here, dR = 1⊙R denotes a diagonal matrix with the elements of the main diagonal of the estimated covariance matrix R.

[0052] In this case, the optimal regularization parameter in equation 4.7 is λ^2=∑i=1p∑j=1,i≠jpVar(R)ij∑i=1p∑j=1,i≠jp|Rij|2.

[0053] The optimized estimators S1 and S2 for the covariance matrix thus generated can, like any two other estimators for the covariance matrix, be S=αS1+(1−α)S2 using a parameter α ∈ ℝ, also called the first weighting factor. In this way, the MSE will be further reduced for reasonable estimators S1 and S2. In this context, "reasonable" means that combining two estimators can only be helpful if both inherently have a sufficiently small MSE.

[0054] When combining estimators according to Equation 4.19, it is sufficient to use a regularized estimator to make the result nonsingular if all other estimators are at least positive semidefinite. This is because the regularized estimator leads to a positive definite matrix, and adding further positive semidefinite matrices still yields a positive definite matrix. Thus, the result remains nonsingular.

[0055] If more than two estimators are to be combined, they can be combined sequentially using equation 4.19. When combining different estimators S i the corresponding shrinkage coefficients λ i , for i ∈ ℕ, are assumed to be constant. That is, the regularization parameters λ i are determined for the individual estimators and then these values ​​are used to combine the individual estimators according to equation 4.19.

[0056] For the combination of two estimators according to equation 4.19, the corresponding parameter α can be determined using the shrinkage approach according to equation 4.7. The optimal parameter is accordingly given by α^=∑i=1p∑j=1p(Var(S2)−Cov(S1,S2)−Bias(S2)E{S1−S2})ijE{‖S1−S2‖F2} given.

[0057] This parameter is calculated in detail for the two estimators of the covariance matrix from sections 4.2.1 and 4.2.2 in the following section.

[0058] In the following, the individual components in equation 4.20 are evaluated for the use of the estimators presented above. Firstly, the estimator S1 of the diagonal loading approach is calculated in the form S1=λ^1μ^1+(1−λ^1)R according to equation 4.11. Here, λ ̂ 1 the optimal regularization parameter according to equation 4.16 and µ̂ is chosen according to equation 4.15. On the other hand, the estimator S2 of the diagonal shrinkage approach is given in the form S2=λ^2dR+(1−λ^2)R according to equation 4.17. Here, λ ̂ 2 the optimal value of λ2 according to equation 4.18. λ ̂ 2 is also called the second weighting factor

[0059] In the following, the numerator of equation 4.20 is examined in detail before the denominator is analyzed.

[0060] For the numerator of equation 4.20, the individual terms for using the estimators in equation 4.21 and equation 4.22 are first specified before they are added.

[0061] The variance of the estimator S2 is obtained by inserting: Var(S2)=Var(λ^2dR+(1−λ^2)R)

[0062] Since the variance is given by Var(X) = E{(X - E{X}) 2} is given, Var(S2) can also be written as Var(S2)=E{S2⊙S2}−E{S2}⊙E{S2}.

[0063] For the first term in equation 4.24, by inserting and taking into account that R⊙dR=dR⊙R and R⊙dR=dR⊙R the expression E{S2⊙S2}=(2λ^2−λ22)⋅E{dR⊙dR}+(1−λ2)2E{R⊙R}.

[0064] For the second term in equation 4.24, using E{S2}=λ^2E{dR}−(1−λ^2)E{R} by multiplication E{S2}⊙E{S2}=λ^2E{dR}⊙E{dR}+2λ^2(1−λ^2)E{dR}⊙E{R}+(1−λ^2)2E{R}⊙E{R}.

[0065] By inserting equation 4.27 and equation 4.29 into equation 4.24, the total is Var(S2)=λ^22Var(dR)+(1−λ^2)2Var(R)+2λ^2(1−λ^2)Cov(R,dR).

[0066] Since dR = 1⊙R, the covariance in equation 4.30 can also be represented as a variance. Therefore, ultimately, Var(S2)=λ^2(2−λ^2)Var(dR)+(1−λ^2)2Var(R).

[0067] Covariance of the estimators S1 and S2: For the second term in the numerator of equation 4.20, inserting the estimators S1 and S2 gives Cov(S1,S2)=Cov(λ^1μ^1+(1−λ^1)R,λ^2dR+(1−λ^2)R).

[0068] By subtracting the constants we get Cov(S1,S2)=(1−λ^1)Cov(R,λ^2dR+(1−λ^1)R).

[0069] The calculation of the covariance can be divided into two covariances for the two components of the right-hand term: Cov(S1,S1)=(1−λ^1)λ^2Cov(R,dR)+(1−λ^1)(1−λ^2)Cov(R,R).

[0070] Since the covariance of two equal terms equals the variance, the term Cov(S1,S1)=(1−λ^1)λ^2Var(dR)+(1−λ^1)(1−λ^2)Var(R).

[0071] The bias of the second estimator is obtained by inserting S2: Bias(S2)=E{λ^2dR+(1−λ^2)R−E{R}}.

[0072] The expected value of a sum is equal to the sum of the corresponding expected values, so equation 4.36 can be further simplified: Bias(S2)=E{λ^2dR}+E{(1−λ^2)R}−E{R}}.

[0073] Here, the constant scalars can be extracted from the expected values, so that the term Bias(S2)=λ^2E{dR}−λ^2E{R}=λ^2(E{dR}−E{R}) remains.

[0074] For the expected value of the difference between S1 and S2 I get it by inserting E{S1−S2}=E{λ^1μ^1+(1−λ^1)R−λ^2dR−(1−λ^2)R}.

[0075] By extracting the constant scalars we finally get E{S1−S2}=λ^1μ^1+(λ^2−λ^1)E{R}−λ^2E{dR}.

[0076] To calculate the numerator in equation 4.20, the results from equation 4.31, equation 4.35, equation 4.38 and equation 4.40 are combined accordingly. This results in the expression for the bracket in the numerator of equation 4.20: Z=λ^2(2−λ^2)Var(dR)−(1−λ^1)λ^2Var(dR)+(1−λ^2)2Var(R)−(1−λ^2)(1−λ^1)Var(R) +λ^2(E{R}−E{dR})⊙[λ^1μ1+(λ^2−λ^1)E{R}−λ^2E{dR}].

[0077] Here, ⊙ denotes component-wise or point-wise multiplication. The main diagonal elements of (E{R} - E{dR}) are zero, while all off-diagonal elements of 1 and dR are identically zero. For this reason, the part (E{R}−E{dR})⊙1=0 and (E{R−E{dR}})⊙E{dR}=0.

[0078] This gives the term in equation 4.41 with pointwise multiplication λ^2(E{R}−E{dR})⊙[λ^1μ1+(λ^2−λ^1)E{R}−λ^2E{dR}]=λ^2(λ^2−λ^1)(E{R}−E{dR})⊙E{R}.

[0079] Based on equation 4.43, the right multiplier in equation 4.44 can be extended by E{dR}, so that λ^2(λ^2−λ^1)(E{R}−E{dR})⊙E{R}=λ^2(λ^2−λ^1)(E{R}−E{dR})⊙(E{R}−E{dR})

[0080] With λ^2(2−λ^2)−(1−λ^1)λ^2=λ^2(1+λ^1−λ^2) and (1−λ^2)2−(1−λ^1)(1−λ^2)=(1−λ^2)(1−λ^2−1+λ^1)=(1−λ^2)(λ^1−λ^2) results from equation 4.41 Z=(1+λ^1−λ^2)λ^2Var(dR)+(1−λ^2)(λ^1−λ^2)Var(R) +λ^2(λ^2−λ^1)(E{R}−E{dR})⊙(E{R}−E{dR}).

[0081] The analysis of the denominator of Equation 4.20 is carried out in two parts. First, the Frobenius norm is calculated for the two estimators of the covariance matrix according to Equations 4.21 and 4.22. The result is then divided into diagonal and off-diagonal elements.

[0082] First, the two estimators of the covariance matrix according to equation 4.21 and equation 4.22 are inserted into the Frobenius norm: E{‖S^1−S^2‖F2}=E{‖λ^1μ1+(1−λ^1)R−λ^2dR−(1−λ^2)R‖F2}.

[0083] By combining the terms with R we get E{‖S^1−S^2‖F2}=E{‖λ^1μ^1+(λ^2−λ^1)R−λ^2dR‖F2}.

[0084] The term in equation 4.50 can be split into diagonal and off-diagonal elements. To do this, the first step is to estimate the covariance in R=(R−dR)+dR The first term represents the elements on the secondary diagonals, and the term dR contains the diagonal elements. This gives us from equation 4.50: E{‖S^1−S^2‖F2}=E{‖λ^1μ^1+(λ^2−λ^1)R−λ^2dR‖F2}=E{‖λ^1μ1+λ^2(R−dR)−λ1R‖F2}.

[0085] If this is done with λ ̂ 1dR - λ ̂ 1dR = 0, the result can be E{‖S^1−S^2‖F2}=E{‖λ^1μ^1+λ^1dR+λ^2(R−dR)−λ^1(R−dR)‖F2} By combining the secondary diagonal elements, we ultimately get E{‖S^1−S^2‖F2}=E{‖λ^1(μ^1−dR)+(λ^2−λ^1)(R−dR)‖F2}

[0086] By defining the Frobenius norm according to equation 4.5, the absolute value square is calculated point by point when evaluating equation 4.54, which also allows the Frobenius norm to be divided into diagonal and off-diagonal elements: E{‖S^1−S^2‖F2}=E{‖λ^1(μ^1−dR)‖F2+‖(λ^2−λ^1)(R−dR)‖F2}.

[0087] In this way, the expected value of the Frobenius norm for equation 4.20 can be calculated separately for diagonal elements and off-diagonal elements, which simplifies the implementation of the combination of two estimators because in this way the diagonal and off-diagonal elements of the covariance matrix are used only at one point each.

[0088] A method for determining the optimized covariance matrix using two optimization methods can therefore (sequentially) comprise the following steps: 1) The empirical covariance matrix is ​​determined from the input data using equation 4.2. 2) For the estimation of the covariance matrix using diagonal loading, the parameters µ̂ and λ ̂ 1 calculated using equation 4.15 and equation 4.16. 3) For the estimation of the covariance matrix using the diagonal shrinkage approach, the parameter λ ̂ 2 using equation 4.18. 4) To superimpose both estimators, equation 4.20 is evaluated, using equation 4.48 for the numerator and equation 4.55 for the denominator. 5) Finally, the final estimator of the covariance matrix is ​​calculated using equation 4.19.

[0089] Steps 2) to 4) require knowledge of the variance and expected value of the estimators R and dR. The estimation of these statistical moments will not be explained here, as they are known to those skilled in the art. Several independent empirical covariance matrices are used to estimate these quantities. The estimation of the covariance matrix by averaging the dyadic products of the data vectors is referred to as the empirical covariance matrix.

[0090] This paper presents parameter-free methods for regularized estimation of the covariance matrix from data sets. In this context, parameter-free means that the required parameters can be determined from the (measured) data and do not need to be specified. A practical application is the estimation of covariance matrices for the Minimum Power Distortionless Response (MPDR) beamformer (BF).

[0091] Before determining the empirical covariance matrix, the samples can be normalized. When the empirical covariance matrix is ​​calculated according to equation 4.2, each observation x i , i = 1, ..., N. However, in the case of a non-stationary event, such as a pulse, a few observations have much more energy than others, so that the empirical covariance matrix is ​​dominated by a few observations.

[0092] To ensure that each observation has the same influence on the covariance matrix, a normalization can be introduced. For this purpose, the observations X i = [x i,1 , ..., x i,N ] with the help of x˜i=xi∑j=1p|xi,j| Then, R=1N−1∑i=1N(x˜i−x˜¯)(x˜i−x˜¯)' and the new mean x˜¯=1N∑i=1Nx˜i the covariance matrix is ​​calculated.

[0093] Before normalization, the time signal can be further multiplied by an improved window, for example the von-Hann window (also just “Hann” window), to perform the time-frequency transformation.

[0094] Although some aspects have been described in connection with a device, it is understood that these aspects also represent a description of the corresponding method, so that a block or component of a device can also be understood as a corresponding method step or as a feature of a method step. Similarly, aspects described in connection with or as a method step also represent a description of a corresponding block, detail, or feature of a corresponding device.

[0095] Depending on specific implementation requirements, embodiments of the invention may be implemented in hardware or software. The implementation may be performed using a digital storage medium, such as a floppy disk, a DVD, a Blu-ray Disc, a CD, a ROM, a PROM, an EPROM, an EEPROM, or a FLASH memory, a hard disk, or other magnetic or optical storage device storing electronically readable control signals that can interact or cooperate with a programmable computer system to perform the respective method. Therefore, the digital storage medium may be computer-readable.Some embodiments according to the invention thus comprise a data carrier having electronically readable control signals capable of interacting with a programmable computer system such that one of the methods described herein is carried out.

[0096] In general, embodiments of the present invention can be implemented as a computer program product with a program code, wherein the program code is effective to perform one of the methods when the computer program product is run on a computer. The program code can also be stored, for example, on a machine-readable medium. Other embodiments include the computer program for performing one of the methods described herein, wherein the computer program is stored on a machine-readable medium.

[0097] In other words, one embodiment of the method according to the invention is thus a computer program comprising program code for performing one of the methods described herein when the computer program is run on a computer. Another embodiment of the method according to the invention is thus a data carrier (or a digital storage medium or a computer-readable medium) on which the computer program for performing one of the methods described herein is recorded.

[0098] A further embodiment of the method according to the invention is thus a data stream or a sequence of signals that represents the computer program for carrying out one of the methods described herein. The data stream or the sequence of signals can be configured, for example, to be transferred via a data communication connection, for example, via the Internet.

[0099] A further embodiment comprises a processing device, for example a computer or a programmable logic device, which is configured or adapted to carry out one of the methods described herein.

[0100] A further embodiment comprises a computer on which the computer program for performing one of the methods described herein is installed.

[0101] In some embodiments, a programmable logic device (e.g., a field-programmable gate array, an FPGA) may be used to perform some or all of the functionalities of the methods described herein. In some embodiments, a field-programmable gate array may interact with a microprocessor to perform any of the methods described herein. In general, in some embodiments, the methods are performed by any hardware device. This may be general-purpose hardware such as a computer processor (CPU) or method-specific hardware such as an ASIC.

[0102] The above-described embodiments are merely illustrative of the principles of the present invention. It is understood that modifications and variations of the arrangements and details described herein will be apparent to others skilled in the art. Therefore, it is intended that the invention be limited only by the scope of the following claims and not by the specific details presented in the description and explanation of the embodiments herein. List of reference symbols: 20 underwater sound receivers 22 transducers 24 sound waves 26 Signal 28 Signal processor 30 Representation

Claims

[1] Underwater sound receiver (20) with the following features: a plurality of sound transducers (22), each designed to convert incoming sound waves (24) into a signal (26); a signal processor (28) which is designed to generate an empirical covariance matrix based on the signals (26) R=Cov(X)=1N−1∑i=1N(xi−x¯)(xi−x¯)' to determine and to optimize the empirical covariance matrix to obtain an optimized covariance matrix, where the optimized covariance matrix S=αS1 + (1 - α)S2 has a first Estimator S1 = λ1µ1 + (1 - λ1)R with µ̂ = tr(R) / N and λ^1=∑i=1p∑j=1,i≠jpVar(R)ij+∑jVar(R)jj∑i=1p∑j=1,i≠jp|Rij|2+∑j|Rjj−μ^|2 and a second estimator S2 = λ2dR + (1 - λ2)R with λ^2=∑i=1p∑j=1,i≠jpVar(R)ij∑i=1p∑j=1,i≠jp|Rij|2 has, where a^=∑i=1p∑j=1p(Var(S2)−Cov(S1,S2)−Bias(S2)E{S1−S2})ifE{‖S1−S2‖F2} is selected, wherein the signal processor is designed to be used for the counter Z=(1+λ^1−λ^2)λ^2Var(dR)+(1−λ^2)(λ^1−λ^2)Var(R) +λ^2(λ^1−λ^2)(E{R}−E{dR})⊙(E{R}−E{dR}). to evaluate wherein the signal processor (28) is designed to carry out a direction formation of the sound waves (24) based on the optimized covariance matrix. [2] Underwater sound receiver (20) according to claim 1, E{‖S^1−S^2‖F2}=E{‖λ^1(μ^1−dR)‖F2+‖(λ^2−λ^1)(R−dR)‖F2} to evaluate and for the denominator wherein the signal processor (28) is designed to perform a time-frequency transformation of the signals in order to obtain a spectral mapping of the signals and to determine the empirical covariance matrix based on the spectral mapping of the signals; wherein the signal processor (28) is designed to use a window, in particular a Von-Hann window, for the time-frequency transformation, which uses a smaller factor for multiplying with the signal for first sample values, in particular the first 5 sample values, than a Hamming window. [3] Underwater sound receiver (20) according to one of the preceding claims, wherein the signal processor (28) is designed to form the empirical covariance matrix based on samples of the signal or a spectral mapping of the signal, wherein the signal processor is designed to normalize the samples. [4] Underwater sound receiver (20) according to one of the preceding claims, wherein the signal processor (28) is designed to optimize the empirical covariance matrix by means of a weighted combination of a first optimization method and a second optimization method, wherein the weighted combination is weighted by means of a first weighting factor. [5] Underwater sound receiver (20) according to one of the preceding claims, wherein the signal processor (28) is designed to carry out the optimization by means of regularization. [6] Underwater sound receiver (20) according to one of claims 4 to 5, wherein the signal processor (28) is designed to carry out the optimization by means of the first optimization method (S1) by means of diagonal loading of the empirical covariance matrix. [7] Underwater sound receiver (20) according to one of claims 4 or 6, wherein the signal processor (28) is designed to carry out the optimization by means of the second optimization method (S2) by means of diagonal shrinkage of the empirical covariance matrix. [8] Underwater sound receiver (20) according to one of claims 4 to 7, wherein the signal processor (28) is designed to overlay the empirical covariance matrix with the weighted combination of the first optimization method and / or the second optimization method in order to optimize the empirical covariance matrix. [9] Underwater sound receiver (20) according to claim 8, wherein the signal processor (28) is designed to carry out the superposition by means of a second weighting factor, wherein the second weighting factor is limited to a range between 0 and 1, in particular between 0 and 0.8, preferably between 0 and 0.

7. [10] Underwater sound receiver (20) according to one of the preceding claims, wherein the signal processor (28) is configured to output a pictorial representation of the received energy from the sound waves over a direction obtained by means of the direction formation. [11] Underwater sound receiver (20) according to claim 10, wherein the underwater sound receiver comprises an imaging unit 32 configured to display the pictorial representation. [12] Method for operating an underwater sound receiver comprising the following steps: Using a plurality of sound transducers, each configured to convert incoming sound waves into a signal; Determining an empirical covariance matrix based on the signals; Optimizing the empirical covariance matrix to obtain an optimized covariance matrix, where the optimized covariance matrix S=αS1 + (1- α)S2 has a first Estimator S1 = λ1µ1 + (1- λ1)R with µ̂ = tr(R) / N and λ^1=∑i=1p∑j=1,i≠jpVar(R)ij+∑jVar(R)jj∑i=1p∑j=1,i≠jp|Rij|2+∑j|Rjj−μ^|2 and a second estimator S2 = λ2dR+(1 -λ2)R with λ^2=∑i=1p∑j=1,i≠jpVar(R)ij∑i=1p∑j=1,i≠jp|Rij|2 has, where a^=∑i=1p∑j=1p(Var(S2)−Cov(S1,S2)−Bias(S2)E{S1−S2})E{‖S1−S2‖F2} is selected, whereby for the counter Z=(1+λ^1−λ^2)λ^2Var(dR)+(1−λ^2)(λ^1−λ^2)Var(R) +λ^2(λ^1−λ^2)(E{R}−E{dR})⊙(E{R}−E{dR}) and for the denominator E{‖S^1−S^2‖F2}=E{‖λ^1(μ^1−dR)‖F2+‖(λ^2−λ^1)(R−dR)‖F2} is evaluated; Performing a directionalization of the signals based on the optimized covariance matrix.

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