High-Resolution Signal and Noise Field Estimation Using the&amp;lt;tex&amp;gt;L&amp;lt;/tex&amp;gt;1 (Least Absolute Values) Norm

Zala, Cedric A.; Barrodale, I.; Kennedy, Joe

doi:10.1109/joe.1987.1145232

Cited by 21 publications

(12 citation statements)

References 9 publications

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“…Using Lemma 4 and the matrix norm , we have that Now (31) proving the first inequality. To prove the second one we first note that by the singular value decomposition of , we have (32) where in the last inequality we used (31). In [46] it is shown that where and .…”

Section: Appendix IV Proof Of Theoremmentioning

confidence: 85%

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An $L_1$-Method for the Design of Linear-Phase FIR Digital Filters

Grossmann

Eldar

2007

IEEE Trans. Signal Process.

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Abstract-This paper considers the design of linear-phase finite impulse response digital filters using an 1 optimality criterion. The motivation for using such filters as well as a mathematical framework for their design is introduced. It is shown that 1 filters possess flat passbands and stopbands while keeping the transition band comparable to that of least-squares filters. The uniqueness of 1 -based filters is explored, and an alternation type theorem for the optimal frequency response is derived. An efficient algorithm for calculating the optimal filter coefficients is proposed, which may be viewed as the analogue of the celebrated Remez exchange method. A comparison with other design techniques is made, demonstrating that the 1 approach may be a good alternative in several applications.

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Section: Appendix IV Proof Of Theoremmentioning

confidence: 85%

“…Typically problems are solved by replacing the original continuous problem by a discrete one, and then using linear programming techniques [27], [30], [31]. Discretization techniques are also common for solving the corresponding and measures [32].…”

Section: A Characterization Of the Optimal Filtermentioning

confidence: 99%

An $L_1$-Method for the Design of Linear-Phase FIR Digital Filters

Grossmann

Eldar

2007

IEEE Trans. Signal Process.

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“…This LAD problem is formulated as (3) Using Proposition 1, we turn the problem described in (3) into another form, which is easier to solve.…”

Section: A Theories Of An Nn Of the Lad With Equality Constraintsmentioning

confidence: 99%

Constrained Least Absolute Deviation Neural Networks

Wang

Peterson

2008

IEEE Trans. Neural Netw.

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It is well known that least absolute deviation (LAD) criterion or L 1 -norm used for estimation of parameters is characterized by robustness, i.e., the estimated parameters are totally resistant (insensitive) to large changes in the sampled data. This is an extremely useful feature, especially, when the sampled data are known to be contaminated by occasionally occurring outliers or by spiky noise. In our previous works, we have proposed the least absolute deviation neural network (LADNN) to solve unconstrained LAD problems. The theoretical proofs and numerical simulations have shown that the LADNN is Lyapunov-stable and it can globally converge to the exact solution to a given unconstrained LAD problem. We have also demonstrated its excellent application value in timedelay estimation. More generally, a practical LAD application problem may contain some linear constraints, such as a set of equalities and/or inequalities, which is called constrained LAD problem, whereas the unconstrained LAD can be considered as a special form of the constrained LAD. In this paper, we present a new neural network called constrained least absolute deviation neural network (CLADNN) to solve general constrained LAD problems. Theoretical proofs and numerical simulations demonstrate that the proposed CLADNN is Lyapunov stable and globally converges to the exact solution to a given constrained LAD problem, independent of initial values. The numerical simulations have also illustrated that the proposed CLADNN can be used to robustly estimate parameters for nonlinear curve fitting, which is extensively used in signal and image processing.

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“…The LAD estimation approach has been used in signal processing [4], image restoration [32], and blind source separation [36]. In this section, we apply the proposed cooperative recurrent neural networks (CRNNs) to parameter estimation of system identification under a class of non-Gaussian noise environments.…”

Section: Application To Parameter Estimationmentioning

confidence: 99%

Cooperative Recurrent Neural Networks for the Constrained $L_{1}$ Estimator

Xia

Kamel²

2007

IEEE Trans. Signal Process.

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The constrained least absolute deviation ( 1 ) estimator is an attractive alternative to both the unconstrained 1 estimator and the least-square estimator. This paper introduces a constrained 1 method and proposes two cooperative recurrent neural networks (CRNNs) for the constrained 1 estimator. Unlike existing cooperative neural networks, the proposed two CRNNs have a novel weighting cooperation scheme to integrate individual neural network information automatically. As a special case, the proposed continuous-time CRNN includes the existing continuoustime neural network for unconstrained 1 estimator. Compared with existing continuous-time neural networks for the constrained 1 estimator, the proposed continuous-time CRNN has a lower model complexity and the finite-time convergence to the exact optimal solution without any additional condition. Furthermore, compared with existing numerical algorithms for the constrained 1 estimator, in addition to a low computational complexity, the proposed two CRNNs are suitable for parallel implementation and can deal with the 1 estimation problem with degeneracy. The proposed two CRNNs are applied to parameter estimation problems under non-Gaussian noise environments. Simulation results demonstrate that the proposed CRNNs are indeed effective in dealing with the 1 estimation problem with nonunique solutions and in obtaining a better solution than relevant algorithms.Index Terms-Constrained least absolute deviation (LAD) estimation, cooperative recurrent neural networks (CRNNs), global convergence, linear parameter estimation.

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High-Resolution Signal and Noise Field Estimation Using the<tex>L</tex>1 (Least Absolute Values) Norm

Cited by 21 publications

References 9 publications

An $L_1$-Method for the Design of Linear-Phase FIR Digital Filters

An $L_1$-Method for the Design of Linear-Phase FIR Digital Filters

Constrained Least Absolute Deviation Neural Networks

Cooperative Recurrent Neural Networks for the Constrained $L_{1}$ Estimator

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