# A hybrid Fourier-Prony method

## TL;DR

This paper introduces a hybrid Fourier-Prony method that enhances sparse signal analysis by combining FFT efficiency with Prony's sparsity exploitation, reducing sample requirements and computational cost while maintaining frequency resolution.

## Contribution

The paper presents a novel hybrid algorithm integrating Prony's method with Fourier analysis, improving efficiency for sparse signal processing.

## Key findings

- Achieves same frequency resolution as FFT
- Uses fewer samples than traditional FFT
- Suitable for parallel implementation

## Abstract

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.