Numerical Stability of DFT Computation for Signals with Structured Support

We consider the problem of building numerically stable algorithms for computing Discrete Fourier Transform (DFT) of $N$- length signals with known frequency support of size $k$. A typical algorithm, in this case, would involve solving (possibly poorly conditioned) system of equations, causing numerical instability. When $N$ is a power of 2, and the frequency support is a random subset of $\mathbb{Z}_N$, we provide an algorithm that has (a possibly optimal) $O(k \log k)$ complexity to compute the DFT while solving system of equations that are $O(1)$ in size.