Performance Benchmarks for Line Spectral Estimation: Ordered Ziv-Zakai Characterization and Plug-In Amplitude Error Analysis

Abstract

Line spectral estimation (LSE) involves estimating both spectral frequencies and their associated complex amplitudes. Existing Fisher-information-based benchmarks are local and therefore do not capture either the threshold behavior of frequency estimation or the propagation of frequency errors to subsequent amplitude reconstruction. This paper develops explicit performance benchmarks for LSE from two complementary perspectives: ordered frequency estimation and plug-in amplitude reconstruction. On the frequency side, we develop a computable Ziv-Zakai bound (ZZB)-type benchmark under an ordered prior by combining a generalized-likelihood-ratio-test (GLRT)-based surrogate for the unavailable pairwise kernel with an ordered-prior correction. The resulting benchmark recovers the ordered a priori bound at low signal-to-noise ratio (SNR) and the marginalized frequency-side Cramer-Rao bound (CRB) at high SNR. On the amplitude side, we derive a local transfer characterization for the sequential plug-in estimator and obtain a computable benchmark for the induced amplitude error. The resulting framework explicitly characterizes threshold behavior on the frequency side and error propagation on the amplitude side. Numerical results support the proposed benchmarks across different SNR regimes, snapshot numbers, and model orders.