Nonlinear Least-Squares Approach to Frequency Estimation and Detection for Sinusoidal Signals with Arbitrary Envelope

Abstract

In this paper, we consider the problem of estimating the frequency of a sinusoidal signal whose amplitude could be either constant or time-varying. We present a nonlinear least-squares (NLS) approach when the envelope is time-varying. It is shown that the NLS criterion can be explicitly minimized with respect to the signal amplitude and envelope values, leaving out a periodogram-like function whose peak location gives the frequency estimate. This result is a significant generalization of a similar result on the NLS frequency estimation for a sinusoidal signal with constant amplitude. A statistical analysis shows that the NLS frequency estimator is asymptotically statistically efficient. The problem of detecting amplitude time variations is next addressed. A statistical test is formulated, based on the statistics of the difference between two frequency estimates. The test is computationally efficient and yields as a by-product consistent frequency estimates under either hypothesis (i.e., constant or time-varying amplitude). Numerical examples are included to show the performance both in terms of estimation and detection.