Optimized Spectrum Permutation for the Multidimensional Sparse FFT

A multidimensional sparse fast Fourier transform algorithm is introduced via generalizations of key concepts used in the one-dimensional (1-D) sparse Fourier transform algorithm. It is shown that permutation parameters are of key importance and should not be chosen randomly but instead can be optimized. A connection is made between the sparse Fourier transform algorithm and lattice theory, thus establishing a rigorous understanding of the effect of the permutations on the algorithm performance. Lattice theory is then used to optimize the set of parameters to achieve a more robust and better performing algorithm. Other algorithms using pseudorandom spectrum permutation can also benefit from the methods developed in this paper. The contributions address the case of the exact k-sparse Fourier transform but the underlying concepts can be applied to the general case of finding a k-sparse approximation of the Fourier transform of an arbitrary signal. Simulations illustrate the efficiency and accuracy of the proposed algorithm. The optimizations of the parameters and the improved performance are shown in simulations for the 2-D case where worst case and average case peak signal-to-noise ratio (PSNR) improves by several decibels.