We consider the problem of computing the Discrete Fourier transform (DFT) of an N- length signal which has only k non-zero DFT coefficients at known locations. We say that a set of indices is spectral if there exists a DFT submatrix (square) with those columns that is unitary up to scaling. When the DFT support set is spectral and N is a prime power, we prove that this can be done in O(klogk) operations using k samples provided the DFT support. This is a generalization of a similar recent result for the case when N is a power of 2. © 2021 IEEE.