Comparative Assessment of ℓ1 and ℓ2 Minimization Algorithms for Period and Hidden Period Estimation

Abstract

This paper analyzes <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>- and <inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula>-minimization (min) algorithms for estimating the period and hidden periods of a finite-length discrete-time signal. Existing approaches typically use over-complete Ramanujan Periodic Dictionaries (RPDs) constructed from both divisor and non-divisor Ramanujan subspaces, yet there is little guidance on when <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>- or <inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula>-min is preferable. This paper demonstrates that <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-min can provide improved performance over <inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula>-min when the RPD coefficient representation is sufficiently sparse. We show that RPD-FISTA can obtain accurate period estimates with a smaller maximum candidate period <inline-formula> <tex-math notation="LaTeX">$P_{\max }$ </tex-math></inline-formula> than RPD-<inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula> can. We state the standard FISTA objective-gap convergence bound and derive the corresponding RPD-FISTA theoretical iteration scaling and leading-order crossover conditions relative to RPD-<inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula>. Illustrative and matched-<inline-formula> <tex-math notation="LaTeX">$P_{\max }$ </tex-math></inline-formula> experiments show that RPD-FISTA can reduce the time required to obtain a correct period estimate when a substantially smaller <inline-formula> <tex-math notation="LaTeX">$P_{\max }$ </tex-math></inline-formula> is accompanied by sufficiently few iterations. For signals with multiple hidden periods, we propose using RPD-Tail-FISTA, which iteratively refines an estimated dominant support, reduces shrinkage on that support, and retains the <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> penalty on its complement. We further introduce a signal-adaptive procedure that computes the period-strength threshold separately for each observed signal and period-estimation method. The procedure applies to both multiple-hidden-period and single-period estimation. Using this procedure, we evaluate the RPD-based estimators over 250 independent synthetic realizations under Gaussian and uniform periodic-component distributions. RPD-FISTA is generally more robust at low SNRs, whereas RPD-Tail-FISTA provides the highest empirical period and hidden-period recovery at moderate/high SNRs. Under clean synthetic conditions, RPD-Tail-FISTA remains the most accurate. Annotation-based validation using the MIT−BIH Arrhythmia and MIT−BIH Noise Stress Test databases further shows that the sparse RPD-based estimators provide substantially more reliable local RR-period estimates than RPD-<inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula> and exhibit a similar SNR-dependent crossover.