论文标题
从傅立叶系数的稀疏非谐波信号的精确重建
Exact Reconstruction of Sparse Non-Harmonic Signals from Fourier Coefficients
论文作者
论文摘要
在本文中，我们得出了一种用于实际非谐波傅立叶总和的新重建方法，即可以表示为$ f（t）= \ sum_ = \ sum_ {j = 1}^{k}^{k}γ_{ $ a_ {j} \ in {\ mathbb r} $（或$ a_ {j} \ in {\ mathrm i} {\ mathbb r} $）是成对的。我们的方法基于\ cite {nst18}中最近提出的稳定迭代有理近似算法。对于信号重建，我们使用一组$ f $的经典傅立叶系数，以固定的间隔$（0，p）$带$ p> 0 $。即使$ f $的所有条款可能是非 - $ p $ periodic，我们的重建方法最多需要$ 2K+2 $ 2 $傅立叶系数$ c_ {n}（f）$才能恢复$ f $的所有参数。我们表明，在确切数据的情况下，提议的迭代算法最多终止了$ k+1 $步骤。该算法还可以检测到$ f $的数字$ k $，如果$ k $是先验的未知，$ l> 2k+2 $ fourier系数。因此，我们的方法为已知的数值方法提供了一种新的稳定替代方法，用于恢复基于Prony方法的指数总和。 关键字：稀疏指数总和，非谐波傅立叶总和，稀疏非周期信号的重建，理性近似，AAA算法，Barycentric代表，傅立叶系数
In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form $f(t) = \sum_{j=1}^{K} γ_{j} \, \cos(2πa_{j} t + b_{j})$, where the frequency parameters $a_{j} \in {\mathbb R}$ (or $a_{j} \in {\mathrm i} {\mathbb R}$) are pairwise different. Our method is based on the recently proposed stable iterative rational approximation algorithm in \cite{NST18}. For signal reconstruction we use a set of classical Fourier coefficients of $f$ with regard to a fixed interval $(0, P)$ with $P>0$. Even though all terms of $f$ may be non-$P$-periodic, our reconstruction method requires at most $2K+2$ Fourier coefficients $c_{n}(f)$ to recover all parameters of $f$. We show that in the case of exact data, the proposed iterative algorithm terminates after at most $K+1$ steps. The algorithm can also detect the number $K$ of terms of $f$, if $K$ is a priori unknown and $L>2K+2$ Fourier coefficients are available. Therefore our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method. Keywords: sparse exponential sums, non-harmonic Fourier sums, reconstruction of sparse non-periodic signals, rational approximation, AAA algorithm, barycentric representation, Fourier coefficients
