Discrete-Time Minimum-Mean-Square-Error Filtering
Garry Einicke · InTech eBooks · 2012
We are IntechOpen, the world's leading publisher of Open Access books Built by scientists, for scientists 12.2% 176,000 190M TOP 1% 154 6,500 Discrete-Time Minimum-Mean-Square-Error Filtering 25 2 Discrete-Time Minimum-Mean-Square-Error Filtering IntroductionThis chapter reviews the solutions for the discrete-time, linear stationary filtering problems that are attributed to Wiener [1] and Kolmogorov [2].As in the continuous-time case, a model-based approach is employed.Here, a linear model is specified by the coefficients of the input and output difference equations.It is shown that the same coefficients appear in the system's (frequency domain) transfer function.In other words, frequency domain model representations can be written down without background knowledge of z-transforms.In the 1960s and 1970s, continuous-time filters were implemented on analogue computers.This practice has been discontinued for two main reasons.First, analogue multipliers and op amp circuits exhibit poor performance whenever (temperature-sensitive) calibrations become out of date.Second, updated software releases are faster to turn around than hardware design iterations.Continuous-time filters are now routinely implemented using digital computers, provided that the signal sampling rates and data processing rates are sufficiently high.Alternatively, continuous-time model parameters may be converted into discrete-time and differential equations can be transformed into difference equations.The ensuing discrete-time filter solutions are then amenable to more economical implementation, namely, employing relatively lower processing rates.The discrete-time Wiener filtering problem is solved in the frequency domain.Once again, it is shown that the optimum minimum-mean-square-error solution is found by completing the square.The optimum solution is noncausal, which can only be implemented by forward and backward processes.This solution is actually a smoother and the optimum filter is found by taking the causal part.The developments rely on solving a spectral factorisation problem, which requires pole-zero cancellations.Therefore, some pertinent discrete-time concepts are introduced in Section 2.2 prior to deriving the filtering results.The discussion of the prerequisite concepts is comparatively brief since it mirrors the continuous-time material introduced previously.In Section 2.3 it is shown that the structure of the filter solutions is unchanged -only the spectral factors are calculated differently."If we value the pursuit of knowledge, we must be free to follow wherever that search may lead us.The free mind is not a barking dog, to be tethered on a ten foot-chain."Adlai Ewing Stevenson Jr. www.intechopen.com