Abstract
In frequency analysis an often appearing problem is the reconstruction of a signal from given samples. Since the samples are usually noised, pure interpolating approaches are not recommended and appropriate approximation methods are more suitable as they can be interpreted as a kind of denoising. Two approaches are widely used. One uses the reflection coefficients of a finite sequence of Szegő polynomials and the other one the zeros of the so called Prony polynomial. We show that both approaches are closely related. As a kind of inverse problem, it’s not surprising that they have in common that both methods depend very sensitive on sampling errors. We use known properties of the signal to estimate the positions of the zeros of the corresponding Szegő- or Prony-like polynomials and construct adaptive algorithms to calculate these ones. Hereby, we get the corresponding parameters in the exponential parts of the signal, too. Then, the coefficients of the signal (as a linear combination of such exponential functions) can be obtained from a system of linear equations by minimizing the residuals with respect to a suitable norm as a kind of denoising.
Highlights
We consider a signal h with finite bandlength m of the form
If we have too few samples, we notice that an l1-solution is in most cases the sparsest solution [7], i.e. the obtained approximation of the Prony polynomial has in most cases the minimal number of non zero coefficients in its monomial representation, see e.g. [4, 5, 24]
It is difficult to distinguish between zeros on the unit circle and ‘near’ to it, especially for a noised signal
Summary
J=1 with Re j ∈ [− , 0], ≥ 0, Im j ∈ We sketch an algorithm, based on the method given in [42], to calculate an approximate Prony polynomial by using the l1-norm for the overdetermined case (i.e., if we have at least 2m samples). If we have too few samples (i.e. an underdetermined system), we notice that an l1-solution is in most cases the sparsest solution [7], i.e. the obtained approximation of the Prony polynomial has in most cases the minimal number of non zero coefficients in its monomial representation, see e.g. It is difficult to distinguish between zeros on the unit circle and ‘near’ to it, especially for a noised signal This case deserves a closer attention in Sect.
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