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Beampattern synthesis via a matrix approach for signal power estimation
 IEEE Trans. Signal Process
, 2007
"... Abstract—We present new beampattern synthesis approaches based on semidefinite relaxation (SDR) for signal power estimation. The conventional approaches use weight vectors at the array output for beampattern synthesis, which we refer to as the vector approaches (VA). Instead of this, we use weight ..."
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Abstract—We present new beampattern synthesis approaches based on semidefinite relaxation (SDR) for signal power estimation. The conventional approaches use weight vectors at the array output for beampattern synthesis, which we refer to as the vector approaches (VA). Instead of this, we use weight matrices at the array output, which leads to matrix approaches (MA). We consider several versions of MA, including a (data) adaptive MA (AMA), as well as several dataindependent MA designs. For all of these MA designs, globally optimal solutions can be determined efficiently due to the convex optimization formulations obtained by SDR. Numerical examples as well as theoretical evidence are presented to show that the optimal weight matrix obtained via SDR has few dominant eigenvalues, and often only one. When the number of dominant eigenvalues of the optimal weight matrix is equal to one, MA reduces to VA, and the main advantage offered by SDR in this case is to determine the globally optimal solution efficiently. Moreover, we show that the AMA allows for strict control of mainbeam shape and peak sidelobe level while retaining the capability of adaptively nulling strong interferences and jammers. Numerical examples are also used to demonstrate that better beampattern designs can be achieved via the dataindependent MA than via its VA counterpart. Index Terms—Beamforming, beampattern synthesis, convex optimization, mainbeam shape control, sidelobe control. I.
On convex stability domain and optimization of IIR filters
"... We discuss descriptions of convex domains containing Schur polynomials, built around a given Schur polynomial. We show that the domain described by a positive realness constraint always contains the domain characterized by Rouché’s theorem. We also show how to handle computationally the positive r ..."
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We discuss descriptions of convex domains containing Schur polynomials, built around a given Schur polynomial. We show that the domain described by a positive realness constraint always contains the domain characterized by Rouché’s theorem. We also show how to handle computationally the positive realness condition, using semidefinite programming, in the context of designing stable IIR filters. Two recent methods of Lang [4] and Lu et al [6] for optimizing IIR filters according to a leastsquares criterion are modified to incorporate the positive realness condition and shown experimentally to give similar results. 1
High Accuracy Algorithms for the Solutions of Semidefinite Linear Programs
, 2001
"... hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, i ..."
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hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. ii The University of Waterloo requires the signatures of all persons using or photocopying this thesis. Please sign below, and give address and date. iii Abstract We present a new family of search directions and of corresponding algorithms to solve conic linear programs. The implementation is specialized to semidefinite programs but the algorithms described handle both nonnegative orthant and Lorentz cone problems and Cartesian products of these sets. The primary objective is not to develop yet another interiorpoint algorithm with polynomial time complexity. The aim is practical and addresses an often neglected aspect of the current research in the area, accuracy. Secondary goals, tempered by the first, are numerical efficiency and proper handling of sparsity. The main search direction, called GaussNewton, is obtained as a leastsquares solution to the optimality condition of the logbarrier problem. This motivation ensures that the direction is welldefined everywhere and that the underlying Jacobian is wellconditioned under standard assumptions. Moreover, it is invariant under affine transformation of the space and under orthogonal transformation of the constraining cone. The GaussNewton direction, both in the special cases of linear programming and on the central path of semidefinite programs, coincides with the search directions used in practical implementations. Finally, the MonteiroZhang family of search directions can be derived as scaled projections of the GaussNewton direction. iv
1 Multistage IIR Filter Design Using Convex Stability Domains Defined by Positive Realness —revised version—
, 2003
"... In this paper we consider IIR filter design where both magnitude and phase are optimized using a weighted and sampled leastsquares criterion. We propose a new convex stability domain defined by positive realness for ensuring the stability of the filter and adapt the SteiglitzMcBride (SM), GaussNe ..."
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In this paper we consider IIR filter design where both magnitude and phase are optimized using a weighted and sampled leastsquares criterion. We propose a new convex stability domain defined by positive realness for ensuring the stability of the filter and adapt the SteiglitzMcBride (SM), GaussNewton (GN) and classical descent methods to the new stability domain. We show how to describe the stability domain such that the description is suited to semidefinite programming and is implementable exactly; also, we prove that this domain contains the domain given by Rouché’s Theorem, recently proposed in [1]. Finally, we give experimental evidence that the best designs are usually obtained with a multistage algorithm, where the three above methods are used in succession, each one being initialized with the result of the previous and where the positive realness stability domain is used instead of that defined by Rouché’s Theorem.
ON THE OPTIMIZATION OF THE TAYKINGSBURY 2D FILTERBANK
"... The TayKingsbury transformation is a way to build twochannel 2D filterbanks (with quincunx sampling) from 1D prototypes. The original transformation was not optimized, but designed using a windowing approach. We propose a natural method of optimization which takes into account the properties of ..."
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The TayKingsbury transformation is a way to build twochannel 2D filterbanks (with quincunx sampling) from 1D prototypes. The original transformation was not optimized, but designed using a windowing approach. We propose a natural method of optimization which takes into account the properties of the transformation and connects them to multivariable polynomials that are sumofsquares on the unit circle. The outcome is a semidefinite programming problem giving the optimal transformation. Further optimization of the filterbank is possible via a standard lifting scheme. Two examples of design are presented. 1