In this paper we extend the deterministic sublinear FFT algorithm in Plonka et al. (Numer Algorithms 78:133–159, 2018. https://doi.org/10.1007/s11075-017-0370-5) for fast reconstruction of M-sparse vectors \({\mathbf{x}}\) of length \(N= 2^J\), where we assume that all components of the discrete Fourier transform \(\hat{\mathbf{x}}= {\mathbf{F}}_{N} {\mathbf{x}}\) are available. The sparsity of \({\mathbf{x}}\) needs not to be known a priori, but is determined by the algorithm. If the sparsity M is larger than \(2^{J/2}\), then the algorithm turns into a usual FFT algorithm with runtime \({\mathcal O}(N \log N)\). For \(M^{2} < N\), the runtime of the algorithm is \({\mathcal O}(M^2 \, \log N)\). The proposed modifications of the approach in Plonka et al. (2018) lead to a significant improvement of the condition numbers of the Vandermonde matrices which are employed in the iterative reconstruction. Our numerical experiments show that our modification has a huge impact on the stability of the algorithm. While the algorithm in Plonka et al. (2018) starts to be unreliable for \(M>20\) because of numerical instabilities, the modified algorithm is still numerically stable for \(M=200\).

在本文中，我们在Plonka等人中扩展了确定性亚线性FFT算法。（NUMER算法78：133-159，2018年https://doi.org/10.1007/s11075-017-0370-5），用于快速重建中号-sparse矢量\（{\ mathbf {X}} \）长度的\（N = 2 ^ J \），其中我们假设离散傅立叶变换的所有分量\（\ hat {\ mathbf {x}} = {\ mathbf {F}} _ {N} {\ mathbf {x} } \）可用。\（{{\ mathbf {x}} \}的稀疏性不必事先已知，而是由算法确定。如果稀疏度M大于\（2 ^ {J / 2} \），则该算法将变为运行时为\（{\ mathcal O}（N \ log N）\）的常规FFT算法。为了\（M ^ {2} <N \），算法的运行时间为\（{\ mathcal O}（M ^ 2 \，\ log N）\）。在Plonka等人的方法中提出的修改建议。（2018）导致迭代重建中使用的Vandermonde矩阵的条件数有了显着改善。我们的数值实验表明，我们的修改对算法的稳定性有很大的影响。而在Plonka等人的算法中。（2018）由于数值不稳定性而开始对\（M> 20 \）变得不可靠，因此修改后的算法对于\（M = 200 \）仍然在数值上稳定。