Fast Sparse DFT Computation for Arbitrary Length by Circular Convolution

Efficient computation of the Discrete Fourier Transform (DFT) for signals with structured frequency support remains a significant challenge in signal processing. The traditional Fast Fourier Transform (FFT) algorithm achieves O(NlogN) complexity, but substantial improvements are possible for signals with known structured supports. In prior work, Pochimireddy et al. demonstrated that for signals with homogeneous supports and a length N that is a power of a prime, the DFT can be computed in O(klogk) time, where k is the size of the frequency support. However, this work was restricted signals with specific lengths. This paper extends these results by developing algorithms for signals of arbitrary lengths using circular convolution techniques to achieve the same computational efficiency. Our proposed methods enhance the theoretical framework and practical utility of structured DFT computation, extending the range of applicable signal lengths. Numerous examples demonstrate the effectiveness of our approach, paving the way for more efficient signal processing in various modern applications.