Cyclostationary modelling of rotating machine vibration signals

Learning characteristics of stochastic-gradient-descent algorithms: A general study, analysis, and critique

In this paper, an adaptive notch filter is developed (employing a frequency domain and time domain analysis) for the enhancement and tracking of sinusoids in additive noise, colored or white. The notch filter is implemented as a constrained infinite impulse response filter with the constraint enforced by a single parameter termed the debiasing parameter. The resulting notch filter requires few parameters, facilitates the formation of the desired band rejection filter response, and also leads to various useful implementations (cascade, parallel). For the adaptation of the filter coefficients, the stochastic Gauss-Newton algorithm is used. The convergence of this updating procedure is established by studying the associated differential equation. Also, it is shown that the structure present in the problem enables truncation of the gradient, thereby reducing the complexity of adapting the filter coefficients. Simulation results are presented to substantiate the analysis, and to demonstrate the potential of the notch filtering technique.

Adaptive notch filtering for the retrieval of sinusoids in noise

The LMS adaptive filter algorithm requires a priori knowledge of the input power level to select the algorithm gain parameter μ for stability and convergence. Since the input power level is usually one of the statistical unknowns, it is normally estimated from the data prior to beginning the adaptation process. It is then assumed that the estimate is perfect in any subsequent analysis of the LMS algorithm behavior. In this paper, the effects of the power level estimate are incorporated in a data dependent μ that appears explicitly within the algorithm. The transient mean and second-moment behavior of the modified LMS (NLMS) algorithm are evaluated, taking into account the explicit statistical dependence of μ upon the input data. The mean behavior of the algorithm is shown to converge to the Wiener weight. A constant coefficient matrix difference equation is derived for the weight fluctuations about the Wiener weight. The equation is solved for a white data covariance matrix and for the adaptive line enhancer with a single-frequency input in steady state for small μ. Expressions for the misadjustment error are also presented. It is shown for the white data covariance matrix case that the averaging of about ten data samples causes negligible degradation as compared to the LMS algorithm. In the ALE application, the steady-state weight fluctuations are shown to be mode dependent, being largest at the frequency of the input.

Analysis of the normalized LMS algorithm with Gaussian inputs

The adaptive line enhancer (ALE) was first described as a practical technique for separating the periodic from the broad-band components of an input signal and for detecting the presence of a sinusoid in white noise. Subsequent work has shown that this adaptive filtering structure is applicable to spectral estimation, predictive deconvolution, speech processing, interference rejection, and other applications which have historically used matrix inversion or Levinson's algorithm techniques. This paper uses an eigenvalue-eigenvector analysis of the expected ALE impulse response vector to demonstrate properties of the convergent filter and to quantify the convergence time and characteristics of the ALE. In particular the ALE's response to a sinusoid plus white noise input is derived and compared to a computer simulation of the ALE with such an input. The eigenvalue-eigenvector technique is then used to evaluate the ALE's performance as an adaptive prewhitener for autoregressive (AR) models with white observation noise. A method is demonstrated which prevents the problem of spectral estimation bias which usually accrues from the observation noise.

Transient and convergent behavior of the adaptive line enhancer

The steady-state behavior of the adaptive line enhancer (ALE), a new implementation of adaptive filtering that has application in detecting and tracking narrow-band signals in broad-band noise, is analyzed for a stationary input consisting of multiple sinusoids in white noise. It is shown that the steady-state performance of an L-weight ALE for this case can be modeled by the L × L Wiener-Hopf matrix equation and that this matrix equation can be transformed into a set of 2N coupled linear equations, where N is the number of sinusoids. It is also shown that the expected values of the ALE weights in steady state can be written as a sum of sinusoids and that the amplitude of each sinusoid is coupled to that of all other sinusoids by coefficients that approach zero as the number of ALE weights becomes large. The analytical results are compared to experimental results obtained with a hardware implementation of the ALE of variable length (up to 256 weights) and show good agreement. Theoretical expressions for linear predictive spectral estimates are also derived for multiple sinusoids in white noise. Comparisons are made between the magnitude of the discrete Fourier transform of the ALE weights and the linear predictive spectral estimate for two sinusoids in white noise.

D. Chabries

Papers

Adaptive enhancement of multiple sinusoids in uncorrelated noise