The uncertainty principle and a generalization of a theorem of Tao

Murty, M. Ram; Whang, Junho Peter

doi:10.1016/j.laa.2012.02.009

Cited by 8 publications

(8 citation statements)

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“…Proposition 7 of [11] may also be used to prove Theorem 8.1 above. This Proposition 7 of [11] is analogous to Chebotarëv's theorem. Corollary 8.1 Let C be the code with check matrix from C ⊥ = E j1 , E j2 , .…”

Section: Proofmentioning

confidence: 96%

“…It is thus noted that there may exist a number of different error-correcting pairs for the same code. 7,9,11,13,15 so that C is 3-error correcting (when C ⊥ is mds). The following are 3-error-correcting pairs.…”

Section: Solve the System Of Equations By Decodingmentioning

confidence: 99%

“…Then C is a code with distance ≥ 2t + 1 = 7. Actually C ⊥ is an (19, 10,9) code and C is an (19, 9,11) code. So in fact the code C is a 5-error correcting code but we just have a 3-error correcting pair.…”

Section: Illustrative Examples Of Random Selectionmentioning

confidence: 99%

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Solving underdetermined systems with error-correcting codes

Hurley

2017

IJICOT

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In an underdetermined system of equations Ax = y, where A is an m × n matrix, only u of the entries of y with u < m are known. Thus Ejw, called 'measurements', are known for certain j ∈ J ⊂ {0, 1, . . . , m − 1} where {Ei, i = 0, 1, . . . , m − 1} are the rows of A and |J| = u. It is required, if possible, to solve the system uniquely when x has at most t non-zero entries with u ≥ 2t.Here such systems are considered from an error-correcting coding point of view. The unknown x can be shown to be the error vector of a code subject to certain conditions on the rows of the matrix A. This reduces the problem to finding a suitable decoding algorithm which then finds x.Decoding workable algorithms are shown to exist, from which the unknown x may be determined, in cases where the known 2t values are evenly spaced (that is, when the elements of J are in arithmetic progression) for classes of matrices satisfying certain row properties. These cases include Fourier n × n matrices where the arithmetic difference k satisfies gcd(n, k) = 1, and classes of Vandermonde matrices V (x1, x2, . . . , xn) (with xi = 0) with arithmetic difference k where the ratios xi/xj for i = j are not k th roots of unity. The decoding algorithm has complexity O(nt) and in some cases, including the Fourier matrix cases, the complexity is O(t 2 ). Matrices which have the property that the determinant of any square submatrix is non-zero are of particular interest. Randomly choosing rows of such matrices can then give t error-correcting pairs to generate a 'measuring' code C ⊥ = {Ej|j ∈ J} with a decoding algorithm which finds x. This has applications to signal processing and compressed sensing.

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Section: Proofmentioning

confidence: 96%

Section: Solve the System Of Equations By Decodingmentioning

confidence: 99%

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Solving underdetermined systems with error-correcting codes

Hurley

2017

IJICOT

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“…1Further refinements due to Biro-Meshulam-Tao may be found in [Tao05]; see also [MW12] for a recent generalization. where x t ∈ R d and u t ∈ R m , and ( f t ) T −1 t=0 and (g t ) T −1 t=0 are two families of maps such that…”

Section: Theorem 31 (Pontryagin Maximum Principle Under State-actionmentioning

confidence: 99%

Discrete Time Pontryagin Maximum Principle Under State-Action-Frequency Constraints

Paruchuri

Chatterjee

2019

IEEE Trans. Automat. Contr.

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A. We establish a Pontryagin maximum principle for discrete time optimal control problems under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, and c) constraints on the frequency spectrum of the optimal control trajectories. While the first two types of constraints are already included in the existing versions of the Pontryagin maximum principle, it turns out that the third type of constraints cannot be recast in any of the standard forms of the existing results for the original control system. We provide two different proofs of our Pontryagin maximum principle in this article, and include several special cases fine-tuned to control-affine nonlinear and linear system models. In particular, for minimization of quadratic cost functions and linear time invariant control systems, we provide tight conditions under which the optimal controls under frequency constraints are either normal or abnormal.

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“…Remark 8.8. It is natural to extend the technique of Theorem 8.6 to vector-valued versions of recent uncertainty principle inequalities for finite frames [62], graphs [15], and cyclic groups and beyond [81], [66]. 9.…”

Section: Uncertainty Principlesmentioning

confidence: 99%

Frame Multiplication Theory and a Vector-Valued DFT and Ambiguity Function

Andrews

Benedetto

Donatelli

2018

J Fourier Anal Appl

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Vector-valued discrete Fourier transforms (DFTs) and ambiguity functions are defined. The motivation for the definitions is to provide realistic modeling of multi-sensor environments in which a useful time-frequency analysis is essential. The definition of the DFT requires associated uncertainty principle inequalities. The definition of the ambiguity function requires a component that leads to formulating a mathematical theory in which two essential algebraic operations can be made compatible in a natural way. The theory is referred to as frame multiplication theory. These definitions, inequalities, and theory are interdependent, and they are the content of the paper with the centerpiece being frame multiplication theory.The technology underlying frame multiplication theory is the theory of frames, short time Fourier transforms (STFTs), and the representation theory of finite groups. The main results have the following form: frame multiplication exists if and only if the finite frames that arise in the theory are of a certain type, e.g., harmonic frames, or, more generally, group frames.In light of the complexities and the importance of the modeling of time-varying and dynamical systems in the context of effectively analyzing vector-valued multi-sensor environments, the theory of vector-valued DFTs and ambiguity functions must not only be mathematically meaningful, but it must have constructive implementable algorithms, and be computationally viable. This paper presents our vision for resolving these issues, in terms of a significant mathematical theory, and based on the goal of formulating and developing a useful vector-valued theory. FRAME MULTIPLICATION THEORY AND A VECTOR-VALUED DFT AND AMBIGUITY FUNCTION 3 d. The short-time Fourier transform (STFT) of v with respect to a window function wfor a definitive mathematical treatment. Thus, we think of the window w as centered at t, and we havee. A(v, w) and V w (v) can clearly be defined for functions v, w on R d and for other function spaces besides L 2 (R d ). The quantity |V w (v)| is the spectrogram of v, that is so important in power spectrum analysis, see, e.g., [89], [17], [68], [70], [24], [65], [83].Our goals are the following.• Ultimately, we shall establish the theory of vector-valued ambiguity functions of vector-valued functions v on R d in terms of their discrete periodic counterparts on Z/N Z. • To this end, in this paper, we define vector-valued DFTs and discrete periodic vectorvalued ambiguity functions on Z/N Z in a natural way. The STFT is the guide and the theory of frames, especially the theory of DFT, harmonic, and group frames, is the framework (sic) to formulate these goals. The underlying technology that allows us to obtain these goals is frame multiplication theory.1.3. Outline. We begin with an extended exposition on the theory of frames (Section 2). The reason is that frames are essential for our results, and our results are sometimes not conceived in terms of frames. As such, it made sense to add sufficient background material.The vect...

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The uncertainty principle and a generalization of a theorem of Tao

Cited by 8 publications

References 4 publications

Solving underdetermined systems with error-correcting codes

Solving underdetermined systems with error-correcting codes

Discrete Time Pontryagin Maximum Principle Under State-Action-Frequency Constraints

Frame Multiplication Theory and a Vector-Valued DFT and Ambiguity Function

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