Two Efficient Sparse Fourier Algorithms Using the Matrix Pencil Method

Abstract

The research on efficient computation of sparse signals by various Sparse Fast Fourier Transform (sFFT) algorithms has always been a hot topic in the direction of signal processing. The algorithms can decrease the sampling and running complexity by taking advantage of the signal’s inherent characteristics that a large number of signals are sparse in the frequency domain. The sFFT algorithm is generally divided into two stages: the first stage is bucketization. The process is to divide N frequencies into B buckets through the filter. The main filters used are the flat window filter and the aliasing filter. The second stage is the spectrum recovery. The process is to successfully locate the position of the large frequency in each bucket and successfully calculate the amplitude. Among these steps, the most difficult and time-consuming step is to successfully locate the position of the large value frequency. For the sFFT algorithm based on a flat window filter, the voting method used in sFFT1.0 and sFFT2.0 algorithms should use many rounds, so these two algorithms are time-consuming and indeterministic, while the phase estimation method used in sFFT3.0 and sFFT4.0 algorithms has medium robustness. For sFFT algorithms based on an aliasing filter, the Prony method used in sFFT-DT1.0 algorithm is only applicable to noiseless signals, while the enumeration method used in the sFFT-DT2.0 algorithm has high complexity and poor robustness. In view of the performance of the old methods, new and more efficient methods are needed to achieve spectrum recovery. The spectrum restoration can be converted to estimating the complex amplitudes and attenuation coefficient in the model of the sum of complex exponentials. The matrix pencil method is a standard technique for mode frequency identification. Therefore, we propose the sFFT5.0 algorithm and sFFT-DT3.0 algorithm using the matrix pencil method to do spectrum recovery. These two algorithms are low computational complexity and strong robustness and have achieved good results in the actual comparative test.