No abstract

<p>We introduce two FFT-based ESPRIT algorithms for line spectral estimation which have lower time complexities than the original ESPRIT algorithm's O(N^3). The preferred method, named FFT-ESPRIT, can be characterized as being a kernel-based subspace estimator that achieves super-resolution at O(N log N) for frequency estimates. First, we demonstrate two estimations of the signal subspace via an integral transformation on the row space of the data matrix and the data matrix itself. The subspace-based methods are approximate in nature, and yet perturbation bounds reveal a noise regime in which FFT-ESPRIT exceeds ESPRIT's performance. We demonstrate the behavior of the algorithm across different SNR regimes and show that the estimated signal subspace is statistically efficient. Numerical simulations show that FFT-ESPRIT is more robust than the ESPRIT algorithm at the very low SNRs, and has a nearly identical performance as ESPRIT at higher SNRs.</p>

<p>This paper introduces the problem formulation of kernel-based subspace estimates for line spectral estimation. Subspace methods suffer from cubic time complexity: specifically, the original ESPRIT algorithm relies on obtaining the signal's singular vectors by SVD/EVD. We show that the original ESPRIT algorithm does not need the signal's singular vectors to exploit rotational invariance, but a set of vectors that span the signal model's Vandermonde matrix are sufficient. To exploit this, we introduce two FFT-based ESPRIT algorithms which have lower time complexities than the original ESPRIT algorithm. These fast algorithms rely on the usage of the DFT kernel, i.e.~a Fourier transform on the row space of the data (Hankel) matrix. The preferred method, named FFT-ESPRIT, achieves quasi-linear time complexity due to a fast Hankel matrix-Kernel matrix product algorithm and extensive usage of the FFT. The kernel-based subspace estimates are approximate in nature, and yet perturbation bounds reveal a noise regime in which FFT-ESPRIT exceeds ESPRIT's performance. We demonstrate the behavior of the algorithm across different SNR and super-resolution regimes and show that the estimated signal subspace is statistically efficient. Numerical simulations show that FFT-ESPRIT is more robust than the ESPRIT algorithm at the very low SNRs, and has a nearly identical performance as ESPRIT at higher SNRs. </p>

<p>We introduce two FFT-based ESPRIT algorithms for line spectral estimation which have lower time complexities than the original ESPRIT algorithm's O(N^3). The preferred method, named FFT-ESPRIT, can be characterized as being a kernel-based subspace estimator that achieves super-resolution at O(N log N) for frequency estimates. First, we demonstrate two estimations of the signal subspace via an integral transformation on the row space of the data matrix and the data matrix itself. The subspace-based methods are approximate in nature, and yet perturbation bounds reveal a noise regime in which FFT-ESPRIT exceeds ESPRIT's performance. We demonstrate the behavior of the algorithm across different SNR regimes and show that the estimated signal subspace is statistically efficient. Numerical simulations show that FFT-ESPRIT is more robust than the ESPRIT algorithm at the very low SNRs, and has a nearly identical performance as ESPRIT at higher SNRs.</p>