## Blind Beamforming for Non Gaussian Signals (1993)

Venue: | IEE Proceedings-F |

Citations: | 491 - 31 self |

### BibTeX

@ARTICLE{Cardoso93blindbeamforming,

author = {Jean-François Cardoso and Antoine Souloumiac},

title = {Blind Beamforming for Non Gaussian Signals},

journal = {IEE Proceedings-F},

year = {1993},

volume = {140},

pages = {362--370}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper considers an application of blind identification to beamforming. The key point is to use estimates of directional vectors rather than resorting to their hypothesized value. By using estimates of the directional vectors obtained via blind identification i.e. without knowing the arrray manifold, beamforming is made robust with respect to array deformations, distortion of the wave front, pointing errors, etc ... so that neither array calibration nor physical modeling are necessary. Rather surprisingly, `blind beamformers' may outperform `informed beamformers' in a plausible range of parameters, even when the array is perfectly known to the informed beamformer. The key assumption blind identification relies on is the statistical independence of the sources, which we exploit using fourth-order cumulants. A computationally efficient technique is presented for the blind estimation of directional vectors, based on joint diagonalization of 4th-order cumulant matrices

### Citations

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Citation Context ...1 Table 2: Entrywise and columnwise RMS error in A in a 4 \Theta 3 example. JADE algorithm with T = 700 samples and binary sources. Appendix A: A joint diagonalization algorithm. The Jacobi technique =-=[29]-=- for diagonalizing a unique hermitian matrix is extended for the joint approximate diagonalization of a set N = fN r j1srssg of arbitrary n \Theta n matrices. It consists in minimizing the diagonaliza... |

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Citation Context ...he purely spatial problem. For instance, the blind deconvolution techniques in [15] closely parallels the unitary maximization of (17) or the `reverse criterion' of [10]. Similarly, the CMA algorithm =-=[16]-=- may be implemented in a spatial version [17]. Blind identification may be based on higher-order cumulants only (hence without second-order prewhitening), with the benefit that consistent estimation i... |

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Citation Context ... context, the signal emitted by a spatially coherent source may be estimated by forming the inner product between the array output and a m \Theta 1 vector acting as a spatial filter. The review paper =-=[1]-=- is a good introduction to various strategies for designing spatial filters or `beamformers'. Denote f p the spatial filter designed to extract s p (t), the signal of interest. The simplest approach t... |

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Citation Context ...t the other extreme, stand purely temporal mixtures and the blind deconvolution problem, showing a structure similar to the purely spatial problem. For instance, the blind deconvolution techniques in =-=[15]-=- closely parallels the unitary maximization of (17) or the `reverse criterion' of [10]. Similarly, the CMA algorithm [16] may be implemented in a spatial version [17]. Blind identification may be base... |

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Citation Context ...f its output. A seminal paper is [23], which deals with real signals; see also [24, 25]. For i.i.d. source signals with known, differentiable probability densities, the maximum likelihood approach of =-=[26]-=- provides asymptotically optimal estimates in the noiseless case. Finally, simple solutions can also be implemented if the model (1,2) holds with temporally correlated source signals, in which case no... |

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Citation Context ...itional a priori information, matrix A can be at best identified up to permutation and scaling of its columns. More general considerations on blind identifiability and indetermination can be found in =-=[2]-=-. 3 Advantage can be taken of this indetermination to assume, without any loss of generality, that the source signals have unit variance: Efjs p (t)j 2 g = 1 for 1spsn so that the dynamic range of the... |

90 | Source separation using higher order moments
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Citation Context ...the `weighted covariance' Efjzj 2 zz g have the same eigenvectors. Hence U may be identified as the unitary diagonalizer of the latter, i.e. without even computing the full cumulant matrix Q z (I n ) =-=[3]-=-. If some noise is present though, expression (15) must be evaluated, i.e. the corrective term R 2 z +R z T r(R z ) must be subtracted from the weighted covariance as shown in [4] in the real case. Un... |

63 |
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Citation Context ...if some eigenvalues are identical. These eigenvalues, by (14), are k p u p I n u p = k p ju p j 2 = k p so that the case of degeneracy is when some sources have identical kurtosis. It is suggested in =-=[5]-=- to diagonalize a linear combination of cumulant slices that is to diagonalize Q z (M ) for some matrix M . The p-th eigenvalue of Q z (M ) being k p u p Mu p , degeneracy is very unlikely. This appro... |

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Citation Context ... long as V is kept unitary, this is equivalent to maximizing under unitary constraint the criterion [6]: c 0 (V ) def = X i=1;n jCum(e i ; e i ; e i ; e i )j 2 : (17) This criterion first appeared in =-=[7]-=- where it is obtained via a 4th-order Gram-Charlier expansion of the likelihood function. Very interestingly, Comon [6] arrives at the same criterion by a different 6 approach based on contrast functi... |

54 | Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors
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Citation Context ...-order prewhitening), with the benefit that consistent estimation is possible without modeling the spatial structure of the noise as long as it is independent and normally distributed. The references =-=[10, 13, 18, 19, 20]-=- specifically considers the spatial problem. Blind identification of model (1,2) is closely related to the `source separation' problem since the latter consists in finding a `separating matrix' B such... |

50 |
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Citation Context ...arating matrix' B such that the coordinates of Bx(t) are the source signals (up to the usual indeterminations), possibly corrupted by noise. Adaptive solutions may be based on cumulant criteria as in =-=[21, 22, 14]-=-. More generally, statistical independence at the output of a separating matrix (in the noiseless case) may be exploited by adapting B using nonlinear functions of its output. A seminal paper is [23],... |

36 | Iterative techniques for blind source separation using only fourth order cumulants
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Citation Context ...-order prewhitening), with the benefit that consistent estimation is possible without modeling the spatial structure of the noise as long as it is independent and normally distributed. The references =-=[10, 13, 18, 19, 20]-=- specifically considers the spatial problem. Blind identification of model (1,2) is closely related to the `source separation' problem since the latter consists in finding a `separating matrix' B such... |

21 |
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Citation Context ... unstacking as in (21) yields the eigenmatrices. Eigenmatrices inherit the orthonormality property from the eigenvectors. The same results can be arrived at using a Kroneker product formulation as in =-=[9]-=-. The eigen-structure of Q z derives from (13). It is readily checked that the set fu p u q j1sp; qsng verifies the properties of proposition 3. Orthonormality of the matrices in this set stems from U... |

21 |
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Citation Context ...while Q z (u p u q ) = 0 for p 6= q. Hence the spectrum of Q z is made of n(n \Gamma 1) zero eigenvalues and n eigenvalues equal to the kurtosis of the sources, (a similar device has been proposed in =-=[10]-=- for detecting the number of kurtic sources). With the notations of proposition 3 and after ordering the eigenvalues by decreasing order of magnitude, we define the eigen-set of Q z as the matrix set ... |

14 |
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Citation Context ...nt matrix Q z (I n ) [3]. If some noise is present though, expression (15) must be evaluated, i.e. the corrective term R 2 z +R z T r(R z ) must be subtracted from the weighted covariance as shown in =-=[4]-=- in the real case. Unitary diagonalization of Q z (I n ) is not essentially determined though, if some eigenvalues are identical. These eigenvalues, by (14), are k p u p I n u p = k p ju p j 2 = k p s... |

13 |
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Citation Context ...s no longer necessary. The approach of section 3.1 may be followed, by diagonalizing a correlation matrix Efz(t +s)z(t) g rather than a cumulant matrix. This was independently proposed in [24] and in =-=[27]-=-. As with cumulant matrices, indetermination problems may occur and several correlation matrices (i.e. for variouss) may be jointly diagonalized for the sake of robustness as shown in [28]. A necessar... |

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Citation Context ...th-order Gram-Charlier expansion of the likelihood function. Very interestingly, Comon [6] arrives at the same criterion by a different 6 approach based on contrast functions, which is reminiscent of =-=[8]-=-. Comon also describes an algorithm for maximizing (17) via products of Givens rotations. Unfortunately, the Givens angles at each step cannot be obtained in closed form in the complex case. We propos... |

11 | Fourth-order cumulant structure forcing. Application to blind array processing
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Citation Context ...-order prewhitening), with the benefit that consistent estimation is possible without modeling the spatial structure of the noise as long as it is independent and normally distributed. The references =-=[10, 13, 18, 19, 20]-=- specifically considers the spatial problem. Blind identification of model (1,2) is closely related to the `source separation' problem since the latter consists in finding a `separating matrix' B such... |

8 |
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Citation Context ...ind identification techniques designed in the framework of multichannel ARMA modeling could be applied, provided they are extended to the complex case. See for instance the cumulant-based approach in =-=[12, 13]-=- and [14] for an adaptive approach. At the other extreme, stand purely temporal mixtures and the blind deconvolution problem, showing a structure similar to the purely spatial problem. For instance, t... |

6 |
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Citation Context ...e at the output of a separating matrix (in the noiseless case) may be exploited by adapting B using nonlinear functions of its output. A seminal paper is [23], which deals with real signals; see also =-=[24, 25]-=-. For i.i.d. source signals with known, differentiable probability densities, the maximum likelihood approach of [26] provides asymptotically optimal estimates in the noiseless case. Finally, simple s... |

5 |
Adaptive blind source separation for channel spatial equalization
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(Show Context)
Citation Context ...arating matrix' B such that the coordinates of Bx(t) are the source signals (up to the usual indeterminations), possibly corrupted by noise. Adaptive solutions may be based on cumulant criteria as in =-=[21, 22, 14]-=-. More generally, statistical independence at the output of a separating matrix (in the noiseless case) may be exploited by adapting B using nonlinear functions of its output. A seminal paper is [23],... |

3 |
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Citation Context ...ation techniques designed in the framework of multichannel ARMA modeling could be applied, provided they are extended to the complex case. See for instance the cumulant-based approach in [12, 13] and =-=[14]-=- for an adaptive approach. At the other extreme, stand purely temporal mixtures and the blind deconvolution problem, showing a structure similar to the purely spatial problem. For instance, the blind ... |

3 |
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Citation Context ...blind deconvolution techniques in [15] closely parallels the unitary maximization of (17) or the `reverse criterion' of [10]. Similarly, the CMA algorithm [16] may be implemented in a spatial version =-=[17]-=-. Blind identification may be based on higher-order cumulants only (hence without second-order prewhitening), with the benefit that consistent estimation is possible without modeling the spatial struc... |

3 |
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Citation Context ..., 14]. More generally, statistical independence at the output of a separating matrix (in the noiseless case) may be exploited by adapting B using nonlinear functions of its output. A seminal paper is =-=[23]-=-, which deals with real signals; see also [24, 25]. For i.i.d. source signals with known, differentiable probability densities, the maximum likelihood approach of [26] provides asymptotically optimal ... |

2 |
Tensor based independent component analysis
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(Show Context)
Citation Context ...ces will perfectly do, but more efficient implementations can also be devised, by taking into account additional cumulant symmetries or the fact that only the n most significant eigenpairs are needed =-=[11]-=-. Recall that computation of the eigenmatrices may be bypassed if, for simplicity, joint diagonalization is performed on the parallel set N p . An even simpler implementation is to form a set 1 A Matl... |

2 |
Uffelen, "New methods for signal separation
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(Show Context)
Citation Context ...e at the output of a separating matrix (in the noiseless case) may be exploited by adapting B using nonlinear functions of its output. A seminal paper is [23], which deals with real signals; see also =-=[24, 25]-=-. For i.i.d. source signals with known, differentiable probability densities, the maximum likelihood approach of [26] provides asymptotically optimal estimates in the noiseless case. Finally, simple s... |

1 |
A new eigenstructure based parameter estimation of multichannel moving average processes
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(Show Context)
Citation Context ...ind identification techniques designed in the framework of multichannel ARMA modeling could be applied, provided they are extended to the complex case. See for instance the cumulant-based approach in =-=[12, 13]-=- and [14] for an adaptive approach. At the other extreme, stand purely temporal mixtures and the blind deconvolution problem, showing a structure similar to the purely spatial problem. For instance, t... |

1 |
Eric Moulines, "Second-order blind separation of correlated sources
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(Show Context)
Citation Context ...[24] and in [27]. As with cumulant matrices, indetermination problems may occur and several correlation matrices (i.e. for variouss) may be jointly diagonalized for the sake of robustness as shown in =-=[28]-=-. A necessary identifiability condition is that the source signals have different spectra. A safe approach may consist in the joint diagonalization of a set made of cumulant matrices and of correlatio... |