Closed-form unbiased frequency estimation of a noisy sinusoid using notch filters

1285 small parameter " > 0, has been established. This proposition is based on the assumption of the stabilizability of the boundary-layer system. It was also shown that this connection is only one-direction valid, i.e., the controllability of the reduced-order and boundary-layer systems always yields the controllability of the original system, but not vice versa. The criterion of the impulse-free E 0-controllability of the reduced-order system is derived in the terms of an auxiliary gain matrix K(t). The invariance of this criterion to K(t) is shown. Due to the duality, similar results can be obtained for the Euclidean space observability of singularly perturbed linear time-dependent systems with multiple small delay. In this case, the assumption of the stabilizability of the boundary-layer system has to be replaced by the assumption of the detectability of this system. Controllability and time-optimal control of systems with slow and fast modes, " IEEE Trans. Automat. On the connection between non-linear differential-algebraic equations and singularly perturbed control systems in nonstandard form, " IEEE Trans. control of linear singularly perturbed systems with small state delay, " J. , " Controllability of singularly perturbed linear time-dependent systems with small state delay, " Dyna. The infinite time quadratic control problem for linear systems with state and control delays: an evolution equation approach, " SIAM J. Stability and the infinite time quadratic cost problem for linear hereditary differential systems , " SIAM Asymptotic solution of a singularly perturbed set of functional differential equations of Riccati type encountered in the optimal control theory, " Nonlinear Diff. An estimate for the fundamental matrix of singularly perturbed differential-functional equations and some applications, " Diff. Abstract—In this note, the problem of the frequency estimation of a sinu-soid embedded in white noise is considered. The approach used herein is the minimization of the sample variance of the output of constrained notch filters fed by the noisy sinusoid. In particular, this note focuses on closed-form expressions of the frequency estimate, which can be obtained using notch filters having an all-zeros finite-inpulse response (FIR) structure. The results presented in this note are as follows. 1) It is shown that the FIR notch filters obtained from standard second-order infinite-impulse response (IIR) filters are inadequate. 2) A new second-order IIR notch filter is proposed, which provides an unbiased estimate of the frequency. 3) The FIR filter obtained from the new IIR filter provides a closed-form unbiased frequency estimate. 4) …