Unbiased initial phase estimation for real-valued sinusoids with known frequency via spectral leakage compensation

TY - JOUR

T1 - Unbiased initial phase estimation for real-valued sinusoids with known frequency via spectral leakage compensation

AU - Zhao, Zhe

AU - Zhang, Linyue

AU - Zhang, Feng

N1 - Publisher Copyright: © 2025 The Authors

PY - 2026/2

Y1 - 2026/2

N2 - This paper investigates the problem of initial phase estimation for a real-valued sinusoidal signal with known frequency. We analyze the bias of the conventional maximum likelihood estimator (MLE) and show that it primarily arises from spectral leakage in the discrete Fourier transform (DFT). Based on this observation, we propose a novel unbiased estimator that eliminates the influence of spectral leakage, thereby achieving unbiased estimation of the initial phase. From a theoretical perspective, we prove that a statistic related to the proposed unbiased estimator is not complete. As a result, it is not possible to theoretically establish that the proposed estimator is the minimum variance unbiased estimator (MVUE) within the framework of the Lehmann–Scheffé theorem, due to the incompleteness of the statistic. Nevertheless, Monte Carlo simulations are conducted to evaluate the performance of the proposed estimator under various frequencies, initial phases, and signal-to-noise ratio (SNR) conditions. The results show that the proposed method consistently achieves unbiased estimation and yields a variance close to the Cramér–Rao lower bound (CRLB) in all tested scenarios.

AB - This paper investigates the problem of initial phase estimation for a real-valued sinusoidal signal with known frequency. We analyze the bias of the conventional maximum likelihood estimator (MLE) and show that it primarily arises from spectral leakage in the discrete Fourier transform (DFT). Based on this observation, we propose a novel unbiased estimator that eliminates the influence of spectral leakage, thereby achieving unbiased estimation of the initial phase. From a theoretical perspective, we prove that a statistic related to the proposed unbiased estimator is not complete. As a result, it is not possible to theoretically establish that the proposed estimator is the minimum variance unbiased estimator (MVUE) within the framework of the Lehmann–Scheffé theorem, due to the incompleteness of the statistic. Nevertheless, Monte Carlo simulations are conducted to evaluate the performance of the proposed estimator under various frequencies, initial phases, and signal-to-noise ratio (SNR) conditions. The results show that the proposed method consistently achieves unbiased estimation and yields a variance close to the Cramér–Rao lower bound (CRLB) in all tested scenarios.

KW - Complete statistic

KW - Cramér–Rao lower bound

KW - Initial phase estimation

KW - Lehmann–Scheffé theorem

KW - Real sinusoidal signal

KW - Unbiased estimator

UR - https://www.scopus.com/pages/publications/105012757014

U2 - 10.1016/j.sigpro.2025.110227

DO - 10.1016/j.sigpro.2025.110227

M3 - Article

AN - SCOPUS:105012757014

SN - 0165-1684

VL - 239

JO - Signal Processing

JF - Signal Processing

M1 - 110227

ER -