Abstract
In spectral estimation, one has to determine all parameters of an exponential sum for finitely many (noisy) sampled data of this exponential sum. Frequently used methods for spectral estimation are MUSIC (MUltiple SIgnal Classification) and ESPRIT (Estimation of Signal Parameters via Rotational Invariance Technique). For a trigonometric polynomial of large sparsity, we present a new sparse fast Fourier transform by shifted sampling and using MUSIC resp. ESPRIT, where the ESPRIT based method has lower computational cost. Later this technique is extended to a new reconstruction of a multivariate trigonometric polynomial of large sparsity for given (noisy) values sampled on a reconstructing rank-1 lattice. Numerical experiments illustrate the high performance of these procedures.
Highlights
The problem of spectral estimation resp. frequency analysis arises quite often in signal processing, electrical engineering, and mathematical physics and reads as follows: (P1)Recover the positive integerM, the distinct frequencies φj ∈ (−], and the complex coefficients cj = 0 (j = 1, . . . , M) in the exponential sum of sparsity MM h(x) : = cj e2πiφjx (x ∈ R), j=1 (1.1)if noisy sampled data hk: = h(k) + ek (k = 0, . . . , N − 1) with N ≥ 2 M are given, where ek ∈ C are small error terms with |ek| ≤
In a new unified approach to MUSIC and ESPRIT, we show that both methods are based on singular value decomposition (SVD) of a rectangular Hankel matrix of given sampled data
Following question arises: How can one improve the MUSIC and ESPRIT methods for spectral estimation of exponential sums with large sparsity M? In this paper, we show that this is possible by a special divide-and-conquer technique
Summary
The problem of spectral estimation resp. frequency analysis arises quite often in signal processing, electrical engineering, and mathematical physics (see e.g., the books [1, 2] or the survey [3]) and reads as follows:. We present a new efficient spectral estimation with low computational cost for large sparsity M and a moderate number of given samples, if one has to recover a trigonometric polynomial of large sparsity M. This means we specialize the problem (P1). In the case of successful recovery of a sparse multivariate trigonometric polynomial, all frequency vectors are detected without errors
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