A nonlinear filter is proposed for estimating a complex sinusoidal signal and its parameters (frequency, amplitude, and phase) from measurements corrupted by white noise. This filter is derived by applying an extended complex Kalman filter (ECKF) to a nonlinear stochastic system whose state variables are a function of its frequency and a sample of an original signal, and then, proof of the stability is given in the case of a single complex sinusoid. Simulations demonstrate that the proposed nonlinear filter is effective as a method for estimating a single complex sinusoid and its frequency under a low signal-to-noise ratio (SNR). In addition, the effect of the initial condition in the filter on frequency estimation is also discussed.

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A bstract | The fast Kalman lter provides very quick convergence at the computational complexity of the same order that the LMS algorithm requires. Nevertheless, its performance is still unsatisfactory in system identication because the conventional fast Kalman lter fails to track t i m evarying impulse responses of FIR systems. The failure of tracking is due to the absence of system noise in the statespace model to be used. However, according to the derivation of the fast Kalman lter, it is dicult to theoretically in troducethe term of system noise into the algorithm. In this paper, to overcome the diculties, a new fast ltering algorithm, called a fast H 1 lter, is derived based on the H1 theory .

A novel robust estimator is proposed for extracting a single complex sinusoid and its parameter (frequency) from measurements corrupted by white noise. This estimator is called an H1 1 1 sinusoidal estimator (HSE), which is derived by applying an H 1 1 1 filter to a noisy sinusoidal model with the state-space representation. Simulations demonstrate that the HSE is more robust to the nature of observation noise {v v v k k k } than the Kalman sinusoidal estimator (KSE), which is an improved version of the nonlinear filter previously proposed by the author.

Although the backpropagation (BP) scheme is widely used as a learning algorithm for multilayered neural networks, the learning speed of the BP algorithm to obtain acceptable errors is unsatisfactory in spite of some improvements such as introduction of a momentum factor and an adaptive learning rate in the weight adjustment. To solve this problem, a fast learning algorithm based on the extended Kalman filter (EKF) is presented and fortunately its computational complexity has been reduced by some simplifications. In general, however, the Kalman filtering algorithm is well known to be sensitive to the nature of noises which is generally assumed to be Gaussian. In addition, the H(infinity) theory suggests that the maximum energy gain of the Kalman algorithm from disturbances to the estimation error has no upper bound. Therefore, the EKF-based learning algorithms should be improved to enhance the robustness to variations in the initial values of link weights and thresholds as well as to the nature of noises. The paper proposes H(infinity)-learning as a novel learning rule and to derive new globally and locally optimized learning algorithms based on H (infinity)-learning. Their learning behavior is analyzed from various points of view using computer simulations. The derived algorithms are also compared, in performance and computational cost, with the conventional BP and EKF learning algorithms.