Deterministic Sparse Fourier Transform for Continuous Signals with Frequency Gap

The Fourier transform is a fundamental tool in computer science and signal processing. In particular, when the signal is sparse in the frequency domain---having only $k$ distinct frequencies---sparse Fourier transform (SFT) algorithms can recover the signal in a sublinear time (proportional to the sparsity $k$). Most prior research focused on SFT for discrete signals, designing both randomized and deterministic algorithms for one-dimensional and high-dimensional discrete signals. However, SFT for continuous signals (i.e., $x^*(t)=\sum_{j=1}^k v_j e^{2\pi \mathbf{i} f_j t}$ for $t\in [0,T]$) is a more challenging task. The discrete SFT algorithms are not directly applicable to continuous signals due to the sparsity blow-up from the discretization. Prior to this work, there is a randomized algorithm that achieves an $\ell_2$ recovery guarantee in $\widetilde{O}(k\cdot \mathrm{polylog}(F/\eta))$ time, where $F$ is the band-limit of the frequencies and $\eta$ is the frequency gap.Nevertheless, whether we can solve this problem without using randomness remains open. In this work, we address this gap and introducethe first sublinear-time deterministic sparse Fourier transform algorithm in the continuous setting. Specifically, our algorithm uses $\widetilde{O}(k^2 \cdot \mathrm{polylog}(F/\eta))$ samples and $\widetilde{O}(k^2 \cdot \mathrm{polylog}(F/\eta))$ time to reconstruct the on-grid signal with arbitrary noise that satisfies a mild condition. This is the optimal recovery guarantee that can be achieved by any deterministic approach.

Imagine trying to identify the few distinct notes being played in a piece of music while the orchestra is still performing. Traditional tools like the Fast Fourier Transform check every possible note, which is quick but still slower than it needs to be if only a handful of notes are present, and they usually work only after the music has been neatly digitized. We introduced a new shortcut that never rolls the dice and never needs that rigid digital grid. Our method zooms straight in on the real notes of any continuous-time signal, using only a tiny fraction of the usual measurements and without ever missing a note. Because it’s both quick and fail-safe, this technique can make radar imaging, wireless communications, heart-monitor readings, and other real-time systems faster and more reliable, where every microsecond and every lost signal matter.