Abstract
Computing the Sparse Fast Fourier Transform(sFFT) has emerged as a critical topic for a long time. The sFFT algorithms decrease the runtime and sampling complexity by taking advantage of the signal's inherent characteristics that a large number of signals are sparse in the frequency domain. More than ten sFFT algorithms have been proposed, which can be classified into many types according to filter, framework, method of location, method of estimation. In this paper, the technology of these algorithms is completely analyzed in theory. The performance of them is thoroughly tested and verified in practice. The techniques involved in different sFFT algorithms include the following contents: five operations of signal, three methods of frequency bucketization, five methods of location, four methods of estimation, two problems caused by bucketization, three methods to solve these two problems, four algorithmic frameworks. All the above technologies and methods are introduced in detail and examples are given to illustrate the above research. After theoretical research, we make experiments for computing the signals of different SNR, length, sparsity by a standard testing platform and record the run time, percentage of the signal sampled and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ,L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> error with eight different typical sFFT algorithms. The result of experiments satisfies the inferences obtained in theory. It helps us to optimize these algorithms and use them selectively.
Highlights
The Discrete Fourier Transform(DFT) is one of the most important and widely used techniques in mathematical computing
The filtered signal is equal to the product of the original signal and the spike train filter in the time domain, it can be represented by x′ = DLx and it is equivalent to aliasing in the frequency domain
The filtered signal is equal to the convolution of the original signal and the spike train filter in the time domain, it can be represented by x′ = ULx and it is equivalent to subsampled in the frequency domain
Summary
The Discrete Fourier Transform(DFT) is one of the most important and widely used techniques in mathematical computing. The most popular algorithm to compute DFT is the fast Fourier transform(FFT) invented by Cooley and Tukey, which can compute a signal of size N in O(N logN ) time and use O(N ) samples. The performance of the Ann Arbor fast Fourier transform(AAFFT0.5 [1]) algorithm was later improved in the AAFFT0.9 [2], [3] algorithm through the use of unequally-spaced FFTs and binary search technique for spectrum reconstruction. We present an empirical analysis of the performance of the algorithms in theory and in practice
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
