From FFT to sparse FFT: Innovations in efficient signal processing for large sparse data

This paper explores the advancements from the traditional Fast Fourier Transform (FFT) to the Sparse Fast Fourier Transform (sFFT) and their implications for efficient signal processing of large, sparse datasets. FFT has long been a fundamental component in digital signal processing, significantly lowering the runtime of the Discrete Fourier Transform. However, the ingress of big data has necessitated much more efficient algorithms. In contrast, the sFFT exploits the sparsity in the signals themselves to reduce computational demand, and it becomes very efficient. This paper will discuss the theoretical backing of these two developments, FFT and sFFT, and the algorithmic development in both. In addition, it will also discuss the practical applications of both with emphasis on how the latter outperforms the former in large, sparse data. Comparative analysis shows that sFFT has far greater efficiency and noise tolerance, which is of value for network traffic analysis, astrophysical data analysis, and real-time medical imaging. The purpose of this paper is to provide clarity regarding these transformations and their relationship to being paradigms in modern signal analysis.