03-08-2012, 11:04 AM
Issues Related to the Stability Analysis of a Nonstationary Sinusoid Tracking Algorithm
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Objective
Some issues related to the stability analysis of a method of extraction of nonstationary sinusoids developed by A. K. Ziarani will be addressed. In particular, the parameters that govern the accuracy and time to convergence of the algorithm on the sinusoid’s amplitude, frequency and phase will be numerically characterized. Presentation of this characterization as a set of guidelines will enable the selection of optimum parameter values under different circumstances. A correlation between the numerical analysis of governing parameters and the eigenvalues of the linearized system will also be sought.
Motivation
Adaptive Signal Processing
The tracking algorithm under investigation is a form of adaptive signal processing, or an adaptive notch filter. A filter is a system that allows some frequency components of a signal to pass through unchanged but attenuates components of other frequencies. A notch filter in particular either selects or rejects a specific frequency component of a signal. Traditionally, the discrete Fourier transform has been used to represent and isolate the frequency components of signals, but this method is often impractical when the frequencies of the signal are changing in time [ ]. Instead, adaptive filtering and signal processing techniques have been developed to more adequately handle nonstationary signals (signals with time-varying frequencies).
Adaptive signal processing is used in many applications, including the estimation of time varying biomedical signals and the filtering of variable noise from the biomedical signals [ ]. It is also used to manage noise disturbances in power [ ] and control systems [ ]. The audible sound emitted from vibrating equipment can also be mitigated through the use of adaptive filtering [ ]. In general, adaptive signal processing is well-suited to the estimation of signal frequency.
Advantages of Algorithm over Leading Methods
Leading methods of adaptive filtering in addition to the algorithm under discussion include those developed by Regalia and Kalman [1]. The extended Kalman filter estimates the parameters of a linearized model of the process being considered. Two random processes are included in the state equation and output equation that model the noisiness of the input signal. The two random processes are often represented by covariance matrices. The accuracy of the parameter estimation depends largely on modeling the noise correctly, which includes choosing adequate values in the covariance matrices. The selection of initial conditions of the state variables amplitude, phase and frequency also influences the accuracy significantly. In contrast, Ziarani’s algorithm can track sinusoidal components by merely specifying an input model and not assuming a noise model. His algorithm is also less sensitive to noise and the selection of internal parameters and initial conditions .
Importance of Parameter Study
There remains, however, a hindrance to easy application of Ziarani’s algorithm. The accuracy of the estimated parameters and the time the algorithm takes to converge on the estimations are governed largely by three internal parameters, known as μ1, μ2 and μ3. Optimum parameter values have been selected for the pursuit of sinusoids around 60Hz, but to use the algorithm in other situations, the values must again be optimized. Currently, the parameters are chosen intuitively by persons familiar with the algorithm, or by scaling the input signal such that the heuristically found parameters (in the case of 50/60 Hz operation) may be used.
Data Representation
A significant challenge of this project will be to represent the simulation data meaningfully. There are three controlling parameters of the algorithm, two measurable outcomes for each of amplitude, frequency and phase, and many conditions under which the algorithm can be run. How can so many aspects be represented on one plot? Or how can the data be represented efficiently, using few plots but still conveying meaning? Taking the first condition, frequency, perhaps convergence could be plotted against frequency as a family of μ2 curves. Or, convergence verses a μ parameter with a family of frequency curves.
Representation of the data will change as needed throughout the project as the need arises. Ultimately, the findings of this study will be organized as a set of guidelines and graphs that will enable a user of the algorithm to choose the optimum parameter values for their situation. These final graphs will most likely be a composite of the most useful graphs generated during the characterization process