Compressed Single-Tone Frequency Estimation With Unknown Complex Gain: Singular Rates and Global Identifiability

Abstract

Fixed linear compression can preserve local Fisher information for a sinusoid yet destroy global frequency identification, while a dark response can make local estimation nonregular. We study both failures for a single complex tone observed through a fixed complex-linear sketch with unknown nonzero complex gain and pre-sketch white Gaussian noise. Near an isolated analytic dark frequency, we separate radial signal vanishing from optimized projective contact between the two signed frequency branches. When the gain magnitude is constrained to a fixed nondegenerate interval, finite contact is equivalent to local quotient identifiability and minimax consistency. The sharp mean-square-error rate is determined by the sum of the radial and contact orders; infinite contact produces exact local aliases. Globally, we formulate a worst-frequency efficientinformation objective on whitened row spaces. We solve it exactly for every even output rank, prove uniqueness and quantitative rigidity of the symmetric-edge projector, and solve the evenaperture co-rank-one case. The rank-two optimum is aliased, and a winding obstruction gives a positive lower bound on the information price of global identification. From three outputs onward, sketches that are globally identifying with an immersive projective response are open and dense and have zero price at the level of suprema; for every even rank of at least four, the exact optimizer has this property. Thus local information preservation, singular recoverability, and global identifiability obey distinct compression laws within one estimation model.