## Robust minimum variance beamforming (2005)

### Cached

### Download Links

- [www.stanford.edu]
- [stanford.edu]
- [www.stanford.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | IEEE Transactions on Signal Processing |

Citations: | 62 - 10 self |

### BibTeX

@ARTICLE{Lorenz05robustminimum,

author = {Robert G. Lorenz and Stephen P. Boyd},

title = {Robust minimum variance beamforming},

journal = {IEEE Transactions on Signal Processing},

year = {2005},

volume = {53},

pages = {1684--1696}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract—This paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response. Sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold. In our method, uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction. We choose weights that minimize the total weighted power output of the array, subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid. The robust weight selection process can be cast as a second-order cone program that can be solved efficiently using Lagrange multiplier techniques. If the ellipsoid reduces to a single point, the method coincides with Capon’s method. We describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response. We form separate uncertainty ellipsoids for each component in the signal path (e.g., antenna, electronics) and then determine an aggregate uncertainty ellipsoid from these. We give new results for modeling the element-wise products of ellipsoids. We demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples. Index Terms—Ellipsoidal calculus, Hadamard product, robust beamforming, second-order cone programming.

### Citations

4676 |
Matrix Analysis
- Horn, Johnson
- 1986
(Show Context)
Citation Context ... entries. We denote the Hadamard product of vectors and as The Hadamard product of two matrices is similarly denoted and also corresponds to the element-wise product; it enjoys considerable structure =-=[37]-=-. As with other operators, we will consider the Hadamard product operator to have lower precedence than ordinary matrix multiplication. Lemma 3: For any Proof: Direct calculation shows that the , entr... |

770 | Semidefinite Programming
- Vandenberghe, Boyd
- 1996
(Show Context)
Citation Context ...oid, the problem of finding the minimum volume ellipsoid containing the convex hull of be expressed as the following semidefinite program (SDP): can minimize subject to (52) See Vandenberghe and Boyd =-=[33]-=- and Wu and Boyd [34]. The minimum-volume ellipsoid containing is called the LöwnerJohn ellipsoid. Equation (52) is a convex problem in variables and .For full rank (53) with and . The choice of is no... |

526 | Applied numerical linear algebra
- Demmel
- 1997
(Show Context)
Citation Context ...exity of the RMVB Computation We summarize the algorithm below. In parentheses are approximate costs of each of the numbered steps; the actual costs will depend on the implementation and problem size =-=[31]-=-. As in [25], we will consider a flop to be any single floating-point operation. RMVB Computation Given , strictly feasible and . 1) Calculate . 2) Change coordinates. a) Compute Cholesky factorizatio... |

496 |
Constrained Optimization and Lagrange Multiplier Method
- Bertsekas
- 1982
(Show Context)
Citation Context ...r the proper the proper choice of . A. Lagrange Multiplier Methods It is natural to suspect that we may compute the RMVB efficiently using Lagrange multiplier methods. See, for example, [14] and [22]–=-=[26]-=-. Indeed, this is the case. The RMVB is the optimal solution of minimize subject to (24) if we impose the additional constraint that .Wedefine the Lagrangian associated with (24) as (25) where . To ca... |

266 | Robust convex optimization - Ben-Tal, Nemirovski - 1998 |

232 | Adjustable robust solutions of uncertain linear programs - Ben-Tal, Goryashko, et al. - 2004 |

169 | Determinant maximization with linear matrix inequality constraints
- Vandenberghe, Boyd, et al.
- 1998
(Show Context)
Citation Context ...it is guaranteed to cover all the data points used in the description; the MVE is not robust to data outliers. The computation of the covering ellipsoid is relatively complex; see Vandenberghe et al. =-=[35]-=-. In applications where a real-time response is required, the covering ellipsoid calculations may be profitably performed in advance. V. UNCERTAINTY ELLIPSOID CALCULUS Instead of computing ellipsoid d... |

164 |
High-Resolution Frequency-Wavenumber Spectrum .4naiysis
- Capon
- 1969
(Show Context)
Citation Context ...ently received samples of the array output, e.g., The minimum variance beamformer (MVB) is chosen as the optimal solution of (3) minimize subject to (4) This is commonly referred to as Capon’s method =-=[1]-=-. Equation (4) has an analytical solution given by Equation (4) also differs from (2) in that the power expression we are minimizing includes the effect of the desired signal plus noise. The constrain... |

152 | Application of secondorder cone programming
- Lobo, Vandenberghe, et al.
- 1998
(Show Context)
Citation Context ...nd only if it holds for the value of that maximizes , namely, . By the Cauchy-Schwartz inequality, we see that (19) is equivalent to the constraint (21) which is called a second-order cone constraint =-=[13]-=-. We can then express the robust minimum variance beamforming problem (17) as minimize subject to (22) which is a second-order cone program. See [13]–[16]. The subject of robust convex optimization is... |

150 |
Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications
- BEN-TAL, NEMIROVSKI
- 2001
(Show Context)
Citation Context ...ress the robust minimum variance beamforming problem (17) as minimize subject to (22) which is a second-order cone program. See [13]–[16]. The subject of robust convex optimization is covered in [17]–=-=[21]-=-. By assumption, is positive definite, and the constraint in (22) precludes the trivial minimizer of . Hence, this constraint will be tight for any optimal solution, and we may express (22) in terms o... |

146 | Robust Solutions to Least-Squares Problems with Uncertain Data - GHAOUI, LEBRET - 1997 |

93 |
Convex programming with set-inclusive constraints and applications to inexact linear programming
- Soyster
- 1973
(Show Context)
Citation Context ...n express the robust minimum variance beamforming problem (17) as minimize subject to (22) which is a second-order cone program. See [13]–[16]. The subject of robust convex optimization is covered in =-=[17]-=-–[21]. By assumption, is positive definite, and the constraint in (22) precludes the trivial minimizer of . Hence, this constraint will be tight for any optimal solution, and we may express (22) in te... |

83 |
Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem
- Vorobyov, Gershman, et al.
- 2003
(Show Context)
Citation Context ...quadratic program. While the polyhedron approach is less conservative, the size of the description and, hence, the complexity of solving the problem grows with the number of vertices. Vorobyov et al. =-=[9]-=-, [10] have described the use of second-order cone programming for robust beamforming in the case where the uncertainty in the array response is isotropic. In this paper, we consider the case in which... |

63 |
On robust capon beamforming and diagonal loading
- Li, Stoica, et al.
- 2003
(Show Context)
Citation Context ...function using the Capon beamformer provides an accurate power estimate only when the assumed array manifold equals the actual. Prior to publication, we learned of a work similar to ours by Li et al. =-=[32]-=-, in which the authors suggest that our approachs1692 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 5, MAY 2005 can be “modified to eliminate the scaling ambiguity when estimating the power of ... |

54 | U.: Quadratically constrained least squares and quadratic problems - Golub, Matt - 1991 |

47 |
Least squares with a quadratic constraint
- Gander
(Show Context)
Citation Context ...er for the proper the proper choice of . A. Lagrange Multiplier Methods It is natural to suspect that we may compute the RMVB efficiently using Lagrange multiplier methods. See, for example, [14] and =-=[22]-=-–[26]. Indeed, this is the case. The RMVB is the optimal solution of minimize subject to (24) if we impose the additional constraint that .Wedefine the Lagrangian associated with (24) as (25) where . ... |

42 | SDPSOL: A Parser/Solver for Semidefinite Programming and Determinant Maximization Problems with Matrix Structure. User’s Guide, Version Beta
- Wu, Boyd
- 1996
(Show Context)
Citation Context ...inding the minimum volume ellipsoid containing the convex hull of be expressed as the following semidefinite program (SDP): can minimize subject to (52) See Vandenberghe and Boyd [33] and Wu and Boyd =-=[34]-=-. The minimum-volume ellipsoid containing is called the LöwnerJohn ellipsoid. Equation (52) is a convex problem in variables and .For full rank (53) with and . The choice of is not unique; in fact, an... |

37 | Antenna array pattern synthesis via convex optimization
- Lebret, Boyd
- 1997
(Show Context)
Citation Context ...hich is called a second-order cone constraint [13]. We can then express the robust minimum variance beamforming problem (17) as minimize subject to (22) which is a second-order cone program. See [13]–=-=[16]-=-. The subject of robust convex optimization is covered in [17]–[21]. By assumption, is positive definite, and the constraint in (22) precludes the trivial minimizer of . Hence, this constraint will be... |

14 | Numerical electromagnetics code — NEC-4, method of moments, Part I: User’s manual and - Burke - 1992 |

10 | The performance of matched-field beamformers with mediterranean vertical array data
- Krolik
- 1996
(Show Context)
Citation Context ...angles around the nominal look direction. These are known in the literature as point mainbeamsLORENZ AND BOYD: ROBUST MINIMUM VARIANCE BEAMFORMING 1685 constraints or neighboring location constraints =-=[2]-=-. The beamforming problem with point mainbeam constraints can be expressed as minimize subject to (7) where is an matrix of array responses in the constrained directions, and is an vector specifying t... |

10 |
Robust adaptive beamforming in sensor arrays
- Gershman
- 1999
(Show Context)
Citation Context ...0) The parameter penalizes large values of and has the general effect of detuning the beamformer response. The regularized least squares problem (10) has an analytical solution given by (11) Gershman =-=[4]-=- and Johnson and Dudgeon [5] provide a survey of these methods; see also the references contained therein. Similar ideas have been used in adaptive algorithms; see [6]. Beamformers using eigenvalue th... |

7 | Minimum variance beamforming with soft response constraints - Veen - 1991 |

6 |
Relationships between adaptive minimum variance beamforming and optimal source localization
- Harmanci, Tabrikian, et al.
- 2000
(Show Context)
Citation Context ...e also the references contained therein. Similar ideas have been used in adaptive algorithms; see [6]. Beamformers using eigenvalue thresholding methods to achieve robustness have also been used; see =-=[7]-=-. The beamformer is computed according to Capon’s method, using a covariance matrix that has been modified to ensure that no eigenvalue is less than a factor times the largest, where . Specifically, l... |

6 |
Course reader for EE364: Introduction to Convex Optimization with Engineering Applications
- Boyd, Vandenberghe
- 1998
(Show Context)
Citation Context ... beamformer for the proper the proper choice of . A. Lagrange Multiplier Methods It is natural to suspect that we may compute the RMVB efficiently using Lagrange multiplier methods. See, for example, =-=[14]-=- and [22]–[26]. Indeed, this is the case. The RMVB is the optimal solution of minimize subject to (24) if we impose the additional constraint that .Wedefine the Lagrangian associated with (24) as (25)... |

5 | A.: Adjustable Robust - Ben-Tal, Goryashko, et al. - 2004 |

4 |
Adaptive Filter Theory, ser
- Haykin
- 1996
(Show Context)
Citation Context ... solution given by (11) Gershman [4] and Johnson and Dudgeon [5] provide a survey of these methods; see also the references contained therein. Similar ideas have been used in adaptive algorithms; see =-=[6]-=-. Beamformers using eigenvalue thresholding methods to achieve robustness have also been used; see [7]. The beamformer is computed according to Capon’s method, using a covariance matrix that has been ... |

4 |
A new robust beamforming method with antennae calibration erros
- Wu, Zhang
- 1999
(Show Context)
Citation Context ...lection uses the a priori uncertainties in the array manifold in a precise way; the RMVB is guaranteed to satisfy the minimum gain constraint for all values in the uncertainty ellipsoid. Wu and Zhang =-=[8]-=- observe that the array manifold may be described as a polyhedron and that the robust beamforming problem can be cast as a quadratic program. While the polyhedron approach is less conservative, the si... |

4 | Parallel spatial processing: A cure for signal cancellation in adaptive arrays - Su, Shan, et al. - 1986 |

4 | Adaptive filter theory. Prentice Hall information and system sciences series - Haykin - 1996 |

4 | Ellipsoidal Calculus for Estimation and - Kurzhanski, Vályi - 1997 |

2 |
An ellipsoidal approximation to the Hadamard product of elliproids
- Boyd
- 2002
(Show Context)
Citation Context ... second-order cone programming for robust beamforming in the case where the uncertainty in the array response is isotropic. In this paper, we consider the case in which the uncertainty is anisotropic =-=[11]-=-, [12]. We also show how this problem can be solved efficiently in practice. D. Outline of the Paper The rest of this paper is organized as follows. In Section II, we discuss the RMVB. A numerically e... |

2 |
A “best” mismatched filter response for radar clutter discrimination
- Stutt, Spafford
- 1968
(Show Context)
Citation Context ...e equation . Having computed the value of satisfying , the RMVB is computed according to (31) Similar techniques have been used in the design of filters for radar applications; see Stutt and Spafford =-=[27]-=- and Abramovich and Sverdlik [28]. In principle, we could solve for all the roots of (30) and choose the one that results in the smallest objective value and satisfies the constraint , which is assume... |

2 |
Synthesis of a filter which maximizes the signal-to-noise ratio under additional quadratic constraints. Radio
- Abromovich, Sverdlik
- 1970
(Show Context)
Citation Context ...value of satisfying , the RMVB is computed according to (31) Similar techniques have been used in the design of filters for radar applications; see Stutt and Spafford [27] and Abramovich and Sverdlik =-=[28]-=-. In principle, we could solve for all the roots of (30) and choose the one that results in the smallest objective value and satisfies the constraint , which is assumed in (24). In the next section, h... |

1 |
minimum variance beamforming
- “Matched-field
- 1992
(Show Context)
Citation Context ...o reject undesired signals; this is particularly significant for an array with a small number of elements. We may overcome this limitation by using a using a low-rank approximation to the constraints =-=[3]-=-. The best rank approximation to , in a least squares sense, is given by , where is a diagonal matrix consisting of the largest singular values, is a matrix whose columns are the corresponding left si... |

1 |
adaptive beamforming using worst-case performance optimization via second-order cone programming
- “Robust
- 2002
(Show Context)
Citation Context ...atic program. While the polyhedron approach is less conservative, the size of the description and, hence, the complexity of solving the problem grows with the number of vertices. Vorobyov et al. [9], =-=[10]-=- have described the use of second-order cone programming for robust beamforming in the case where the uncertainty in the array response is isotropic. In this paper, we consider the case in which the u... |

1 |
beamforming in GPS arrays
- “Robust
- 2002
(Show Context)
Citation Context ...d-order cone programming for robust beamforming in the case where the uncertainty in the array response is isotropic. In this paper, we consider the case in which the uncertainty is anisotropic [11], =-=[12]-=-. We also show how this problem can be solved efficiently in practice. D. Outline of the Paper The rest of this paper is organized as follows. In Section II, we discuss the RMVB. A numerically efficie... |

1 | Numerical Methods, ser. Automatic Computation - Dahlquist, Björck - 1974 |

1 |
Ellipsoidal Calculus for Estimation and Control, ser. Systems and Control: Foundations and Applications
- Kurzhanski, Vályi
- 1997
(Show Context)
Citation Context ...ix as where is any matrix square root satisfying . Let and . The range of values of the geometrical (or Minkowski) sum is contained in the ellipsoid for all , where (54) (55) see Kurzhanski and Vályi =-=[36]-=-. The value of is commonly chosen to minimize either the determinant or the trace of . Minimizing the trace of in (55) affords two computational advantages over minimizing the determinant. First, comp... |

1 | A comparison of adaptire algorithms based on methods of steepest descent and random search - Widrow, McCool - 1976 |

1 | Course reader for EE36J: Introduction to Convex Optimization with Engineering Applications - Boyd, Vandenberghe - 1999 |