Two-Point Resolution in Spectral Super-Resolution

Two-point super-resolution is an important problem in many signal processing applications. In this paper, we aim to establish a resolution theory for two-point super-resolution from a single snapshot. We consider a complex two-point model with unequal amplitudes and a nontrivial relative phase, and derive super-resolution upper bounds (SRUs) guaranteeing resolvability as well as super-resolution lower bounds (SRLs) below which stable reconstruction is impossible. The resulting bounds provide an explicit characterization of how the amplitude ratio and, more importantly, the relative phase affect the resolution limit for both source-number detection and location estimation. In the in-phase regime, the classical resolution exponents are retained: \((σ/m)^{1/2}\) for source-number detection and \((σ/m)^{1/3}\) for location estimation. In the out-of-phase regimes, the phase term significantly changes the resolution limit: it acts as a direct subtractive term in the near-endpoint regime, and improves the scaling orders in the large-phase regime to \(σ/m\) for source-number detection and \((σ/m)^{1/2}\) for location estimation. Extensive numerical experiments across different phase regimes and reconstruction algorithms validate the predicted scaling laws and theoretical resolution boundaries. Moreover, comparison with our resolution limit in all phase regimes reveals the optimality of \(\ell_0\), ML, and ESPRIT algorithms, and the non-optimality of SVT, MUSIC, and the convex method, a finding that, to the best of our knowledge, has not been reported before. Collectively, our results show that the phase of amplitudes is not merely a nuisance in super-resolution, but a key factor that can be exploited to improve stable resolvability.