A hybrid Fourier-Prony method
This work addresses the need for efficient algorithms for sparse signal representation, offering a faster alternative to FFT for such signals.
The paper proposes a hybrid Fourier-Prony method that combines the discrete Fourier transform with Prony's method to achieve the same frequency resolution as FFT but with fewer samples and lower computational cost, and it is suitable for parallel implementation.
The FFT algorithm that implements the discrete Fourier transform is considered one of the top ten algorithms of the $20$th century. Its main strengths are the low computational cost of $\mathcal{O}(n \log n$) and its stability. It is one of the most commonly used algorithms to analyze signals with a dense frequency representation. In recent years there has been an increasing interest in sparse signal representations and a need for algorithms that exploit such structure. We propose a new technique that combines the properties of the discrete Fourier transform with the sparsity of the signal. This is achieved by integrating ideas of Prony's method into Fourier's method. The resulting technique has the same frequency resolution as the original FFT algorithm but uses fewer samples and can achieve a lower computational cost. Moreover, the proposed algorithm is well suited for a parallel implementation.