Improved bounds for a deterministic sublinear-time Sparse Fourier Algorithm

This paper improves on the best-known runtime and measurement bounds for a recently proposed Deterministic sublinear-time Sparse Fourier Transform algorithm (hereafter called DSFT). In (Iwen, 2008 ), (Iwen, 2007), it is shown that DSFT can exactly reconstruct the Fourier transform (FT) of an N-bandwidth signal f, consisting of B Lt N non-zero frequencies, using O(B²ldrpolylog(N)) time and O(B² ldr polylog(N)) f-samples. DSFT works by taking advantage of natural aliasing phenomena to hash a frequency- sparse signal's FT information modulo O(B ldr polylog(N)) pairwise coprime numbers via O(B ldr polylog(N)) small Discrete Fourier Transforms. Number theoretic arguments then guarantee the original DFT frequencies/coefficients can be recovered via the Chinese Remainder Theorem. DSFT's usage of primes makes its runtime and signal sample requirements highly dependent on the sizes of sums and products of small primes. Our new bounds utilize analytic number theoretic techniques to generate improved (asymptotic) bounds for DSFT. As a result, we provide better bounds for the sampling complexity/number of low-rate analog-to-digital converters (ADCs) required to deterministically recover frequency-sparse wideband signals via DSFT in signal processing applications (Laska, 2006), (Kirolos et al., 2006).