  The Fast Fourier Transform (FFT) is the most efficiently known way to compute
the Discrete Fourier Transform (DFT) of an arbitrary n-length signal, and has a
computational complexity of O(n log n). If the DFT X of the signal x has only k
non-zero coefficients (where k < n), can we do better? In [1], we addressed
this question and presented a novel FFAST (Fast Fourier Aliasing-based Sparse
Transform) algorithm that cleverly induces sparse graph alias codes in the DFT
domain, via a Chinese-Remainder-Theorem (CRT)-guided sub-sampling operation of
the time-domain samples. The resulting sparse graph alias codes are then
exploited to devise a fast and iterative onion-peeling style decoder that
computes an n length DFT of a signal using only O(k) time-domain samples and
O(klog k) computations. The FFAST algorithm is applicable whenever k is
sub-linear in n (i.e. k = o(n)), but is obviously most attractive when k is
much smaller than n.
  In this paper, we adapt the FFAST framework of [1] to the case where the
time-domain samples are corrupted by a white Gaussian noise. In particular, we
show that the extended noise robust algorithm R-FFAST computes an n-length
k-sparse DFT X using O(klog ^3 n) noise-corrupted time-domain samples, in
O(klog^4n) computations, i.e., sub-linear time complexity. While our
theoretical results are for signals with a uniformly random support of the
non-zero DFT coefficients and additive white Gaussian noise, we provide
simulation results which demonstrates that the R-FFAST algorithm performs well
even for signals like MR images, that have an approximately sparse Fourier
spectrum with a non-uniform support for the dominant DFT coefficients.
