The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices can date back to the early twentieth century. It forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vandermonde decomposition. The existence and uniqueness of the decomposition are studied under explicit conditions on the Toeplitz matrix. The new result is connected by duality to the positive real lemma for trigonometric polynomials nonnegative on the same frequency interval. Its applications in the theory of moments and line spectral estimation are illustrated. In particular, it provides a solution to the truncated trigonometric K-moment problem. It is used to derive a primal semidefinite program formulation of the frequency-selective atomic norm in which the frequencies are known a priori to lie in a certain frequency band. Numerical examples are also provided.
The problem of frequency estimation with restriction on the frequency band was studied in @cite_3 @cite_7 @cite_6 . In @cite_3 , an FS atomic norm formulation (or constrained atomic norm in the language of @cite_3 ) was proposed and a dual SDP formulation was presented by applying the theory of positive trigonometric polynomials. In contrast to this, we show in this paper that a primal SDP formulation of the FS atomic norm can be obtained by applying the new FS Vandermonde decomposition. In @cite_7 , the interval prior was interpreted as a prior distribution of the frequencies and a weighted atomic norm approach was then devised that is an approximate but faster implementation of the FS atomic norm. Although the paper @cite_10 does not provide or imply the FS Vandermonde decomposition result, it obtained independently a primal SDP formulation of the FS atomic norm based on a different technique.
@cite_10 @cite_7 @cite_3 @cite_6
2516586151 2099641086 2402002470 2048738839
A classical result in mathematics dating back to the early twentieth century states that a rank-deficient, positive semidefinite, Toeplitz matrix admits a unique Vandermonde decomposition. This forms the basis of modern subspace and recent continuous compressed sensing (or gridless sparse) methods for frequency estimation. Conventionally, the Vandermonde decomposition is defined on the entire frequency domain; but in this paper, we show that the decomposition can be restricted to a prescribed frequency interval if the Toeplitz matrix also satisfies another linear matrix inequality. Besides solving an open classical moment problem in mathematics, this result is applied to practical frequency estimation scenarios in which the frequencies are known a priori to lie in certain frequency intervals. We show that the recent continuous compressed sensing methods derived based on the standard Vandermonde decomposition can be modified in a universal yet simple way to exploit this prior knowledge by applying the new Vandermonde decomposition result. Numerical results are provided to illustrate advantages of the proposed solutions. Suppose a discrete-time signal S(t), 0 spl les t<N, is a superposition of atoms taken from a combined time-frequency dictionary made of spike sequences 1 sub t= spl tau and sinusoids exp 2 spl pi iwt N spl radic N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discrete-time signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time-frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the l sup 1 norm of the coefficients among all decompositions. Here "highly sparse" means that N sub t +N sub w < spl radic N 2 where N sub t is the number of time atoms, N sub w is the number of frequency atoms, and N is the length of the discrete-time signal. Underlying this result is a general l sup 1 uncertainty principle which says that if two bases are mutually incoherent, no nonzero signal can have a sparse representation in both bases simultaneously. For the above setting, the bases are sinusoids and spikes, and mutual incoherence is measured in terms of the largest inner product between different basis elements. The uncertainty principle holds for a variety of interesting basis pairs, not just sinusoids and spikes. The results have idealized applications to band-limited approximation with gross errors, to error-correcting encryption, and to separation of uncoordinated sources. Related phenomena hold for functions of a real variable, with basis pairs such as sinusoids and wavelets, and for functions of two variables, with basis pairs such as wavelets and ridgelets. In these settings, if a function f is representable by a sufficiently sparse superposition of terms taken from both bases, then there is only one such sparse representation; it may be obtained by minimum l sup 1 norm atomic decomposition. The condition "sufficiently sparse" becomes a multiscale condition; for example, that the number of wavelets at level j plus the number of sinusoids in the jth dyadic frequency band are together less than a constant times 2 sup j 2 . This paper concerns the line spectral estimation problem within the recent super-resolution framework. The frequencies of interest are assumed to follow a prior probability distribution. To effectively and efficiently exploit the prior information, we devise a weighted atomic norm approach that is physically sound and can be formulated as convex programming like the standard atomic norm method. Numerical simulations are provided to demonstrate the superior performance of the proposed approach in accuracy and speed compared to the state-of-the-art. Consider a system governed by the time-dependent Schrodinger equation initialized in its ground state. When subjected to weak (size e) parametric forcing by an “ionizing field” (time-varying), the state decays with advancing time due to coupling of the bound state to radiation modes. The decay rate of this metastable state is governed by Fermi’s golden rule, Γ[V], which depends on the potential V and the details of the forcing. We pose the following potential design problem: find Vopt which minimizes Γ[V] (maximizes the lifetime of the state) over an admissible class of potentials with fixed spatial support. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. Then, using quasi-Newton methods, we compute locally optimal potentials. These have the structure of a truncated periodic potential with a localized defect. In contrast to optimal structures for other spectral optimization problems, our optimizing potentials appear to be interior point...