Continuous Fast Fourier Sampling

Fourier sampling algorithms exploit the spectral sparsity of a signal to reconstruct it quickly from a small number of samples. In these algorithms, the sampling rate is subNyquist and the time to reconstruct the dominate frequencies depends on the type of algorithm—some scale with the number of tones found and others with the length of the signal. The Ann Arbor Fast Fourier Transform (AAFFT) scales with the number of desired tones. It approximates the DFT of a spectrally sparse digital signal on a fixed block by taking a small number of structured random samples. Unfortunately, to acquire spectral information on a particular block of interest, the samples acquired must be appropriately correlated for that block. In other words, the sampling pattern, though random, depends on the block of interest. When blocks of interest overlap significantly, the union of the sampling patterns may not be an optimal one (it might not be sub-Nyquist anymore). Unlike the much slower algorithms, the sampling pattern does not accommodate an arbitrary block position. We propose a new sampling procedure called Continuous Fast Fourier Sampling which allows us to continuously sample the signal at a sub-Nyquist rate and then apply AAFFT on any arbitrary block. Thus, we have a highly resource-efficient continuous Fourier sampling algorithm.