{"paper":{"title":"Computing a k-sparse n-length Discrete Fourier Transform using at most 4k samples and O(k log k) complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","cs.MM","math.IT"],"primary_cat":"cs.DS","authors_text":"Kannan Ramchandran, Sameer Pawar","submitted_at":"2013-05-04T02:54:59Z","abstract_excerpt":"Given an $n$-length input signal $\\mbf{x}$, it is well known that its Discrete Fourier Transform (DFT), $\\mbf{X}$, can be computed in $O(n \\log n)$ complexity using a Fast Fourier Transform (FFT). If the spectrum $\\mbf{X}$ is exactly $k$-sparse (where $k<<n$), can we do better? We show that asymptotically in $k$ and $n$, when $k$ is sub-linear in $n$ (precisely, $k \\propto n^{\\delta}$ where $0 < \\delta <1$), and the support of the non-zero DFT coefficients is uniformly random, we can exploit this sparsity in two fundamental ways (i) {\\bf {sample complexity}}: we need only $M=rk$ deterministica"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.0870","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}