Modelling Unknown System - Essay Example

Finite impulse response filters known as Finite Impulse Response are fed forward or nonrecursive filters, which are stable since they have no feedback. Finite impulse response filters can have linear phase characteristic unlike the IIR making them a stable form of filter. However, these filters are not always the desired choice that is why they are facing out on the market. LMS is one of the two basic algorithms in the area of adaptive filtering; however, these algorithms in their simplest forms suffer from several drawbacks and limitations [4].

The convergence of LMS filters is flawed by two main problems: the spread of the eigenvalue correlation matrix of the input signal and the coupling between modes of convergence. Eigenvalue spread results in nonuniform speed of convergence for the filter values; mode coupling results in nonmonotonic trajectories toward convergence of coefficients of filter and in eigenvalue propagation of the disparity effects between the various modes. This leads to irrecoverable instability problems in the finite impulse response filters.

In order to improve on the normal LMS algorithm, alternative adaptive structures like the LMS lattice and the LMS frequency-domain are designed for mode coupling counteraction, though at the price of a greater non adjustment. Pre-whitening filters are proposed applications in system identification and time-delay estimation to reduce the eigenvalue spread consequences [4]. Yule-Walker equations and its mathematics as applied to solving the various problems. The equation is applied in the estimation of the autoregressive (AR) parameters of an observed AR process in time-series analysis, with varied applications that include: blind channel identification, speech analysis, signal detection, spectral estimation, adaptive filtering and speech coding.

Yule-Walker equations are a classical tool for the estimation problem applied to autocorrelation [3]. When the driving noise is Gaussian, the estimate resulting from solving the Yule-Walker equations with the correlations estimated coincides asymptotically. This occurs when the end effects are negligible with the maximum Likelihood (ML) estimate. This estimate is asymptotically unbiased and optimal in the sense of mean square estimation error, asymptotically attaining the Cram?er-Rao lower bound (CRLB) associated with it [3].

However, with non-Gaussian driven noise, the estimate resulting is no longer ML (maximum likelihood estimate) and may be far from the optimal. The derivation and computation of the ML estimate may then become computationally clumsy in some cases. For the case of a Gaussian-Mixture which is intractable, it is of interest, in such cases, to look for other, simpler estimates, which, although not optimal, may still offer significant improvement over the correlations based estimate [3]. Autocorrelation is the similarity between the observations and time of separation between signals.

It is termed as the mathematical tool for determining repetitive patterns like periodic signals damped under noise. It is also used for locating and identifying the missing basic frequency in a signal implied by its harmonic frequencies, often used for processing of signals for analyzing functions [2]. Autocorrelation is used in processing of signal for evaluating the series of values and functions such as time domain signals. Autocorrelation