Reference Work Entry

Encyclopedia of Algorithms

pp 1081-1086

Date:

Learning Significant Fourier Coefficients over Finite Abelian Groups

  • Adi AkaviaAffiliated withDepartment of Electrical Engineering and Computer Science, MIT

Keywords

Finding heavy fourier coefficients Learning heavy fourier coefficients

Years and Authors of Summarized Original Work

  • 2003; Akavia, Goldwasser, Safra

Problem Definition

Fourier transform is among the most widely used tools in computer science. Computing the Fourier transform of a signal of length N may be done in time \( { \Theta(N\log N) } \) using the Fast Fourier Transform (FFT) algorithm. This time bound clearly cannot be improved below \( { \Theta(N) } \), because the output itself is of length N. Nonetheless, it turns out that in many applications it suffices to find only the significant Fourier coefficients, i.e., Fourier coefficients occupying, say, at least \( { 1\,\% } \) of the energy of the signal. This motivates the problem discussed in this entry: the problem of efficiently finding and approximating the significant Fourier coefficients of a given signal (SFT, in short). A naive solution for SFT is to first compute the entire Fourier transform of the given signal and then to output only the significant Fourier coefficients; thus yie ...

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