# Design, analysis and generalization of polynomial predictive filters

##### Koppinen, K. (2003)

Koppinen, K.

Tampere University of Technology

2003

Tietotekniikan osasto - Department of Information Technology

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**Julkaisun pysyvä osoite on**

http://urn.fi/URN:NBN:fi:tty-200810021044

##### Tiivistelmä

New methods for the design, analysis and generalization of polynomial predictive filters (polynomial predictors for short) are developed in this thesis. Polynomial predictors are a subclass of linear mathematical filters, i.e. linear transformations that transform an input sequence of numbers to an output sequence. Filters are in general used to modify or extract useful information from a signal. Examples of filters are echo cancellers in telephone networks, vocal tract filters in speech synthesizers and analysis filters for EEG signals. Polynomial predictors are useful in situations where the signal of interest changes relatively slowly. Examples of slowly varying signals that can be successfully modeled by polynomials include displacement curves of elevators, the received power level in mobile communication systems and angular accelaration in motor drives.

Generally, the filtering of a signal causes delay. However, in many applications, particularly control systems, it is desirable to minimize the delay caused by filtering. Furthermore, the signals in any real-world application contain noise, which we also wish to minimize by filtering. For example, calculating the moving average of a signal generally reduces the noise in the signal, but also causes delay. Polynomial predictors are filters that can predict polynomial signals, and in addition reduce the noise in the signal.

In this thesis we determine the precise class of IIR filters which are polynomial predictors, which unifies the various previously proposed predictor structures. Design methods are developed for optimal least-squares FIR polynomial predictors, which enable arbitrary prediction steps, error weighting and an arbitrary desired frequency response. The parameters of predictors based on the general IIR structure are optimized, yielding improvements over the best previously known IIR polynomial predictors. An analysis of the asymptotic properties of polynomial predictors as the filter length grows without bound is carried out, yielding insight into the characteristics of predictors in general, as well as proving a conjecture on the asymptotic behaviour of the noise gain.

Predictors are generalized for more general signal models than polynomials, and the resulting signal model (corresponding to the solutions of linear homogenous finite difference equations) is shown to be the most general possible. The generalization of the signal model can be interpreted in terms of a z-plane diagram which specifies the complex exponential signals that are predicted. The design methods for FIR polynomial predictors are also carried over to this general signal model. The generalized signal model includes e.g. sinusoidal signals and the design methods are applied to finding improved filters for use in a 50Hz line frequency signal processing application.

In addition to the signal model, the filter type is also generalized. This yields design methods for finding the optimal parameters of differentiators, predictive differentiators and integrators, among others. The asymptotic noise gain of generalized predictors is derived and found to depend on the largest modulus of the complex numbers which specify the signal model. Furthermore, an efficient implementation for generalized FIR predictors is derived.

Generally, the filtering of a signal causes delay. However, in many applications, particularly control systems, it is desirable to minimize the delay caused by filtering. Furthermore, the signals in any real-world application contain noise, which we also wish to minimize by filtering. For example, calculating the moving average of a signal generally reduces the noise in the signal, but also causes delay. Polynomial predictors are filters that can predict polynomial signals, and in addition reduce the noise in the signal.

In this thesis we determine the precise class of IIR filters which are polynomial predictors, which unifies the various previously proposed predictor structures. Design methods are developed for optimal least-squares FIR polynomial predictors, which enable arbitrary prediction steps, error weighting and an arbitrary desired frequency response. The parameters of predictors based on the general IIR structure are optimized, yielding improvements over the best previously known IIR polynomial predictors. An analysis of the asymptotic properties of polynomial predictors as the filter length grows without bound is carried out, yielding insight into the characteristics of predictors in general, as well as proving a conjecture on the asymptotic behaviour of the noise gain.

Predictors are generalized for more general signal models than polynomials, and the resulting signal model (corresponding to the solutions of linear homogenous finite difference equations) is shown to be the most general possible. The generalization of the signal model can be interpreted in terms of a z-plane diagram which specifies the complex exponential signals that are predicted. The design methods for FIR polynomial predictors are also carried over to this general signal model. The generalized signal model includes e.g. sinusoidal signals and the design methods are applied to finding improved filters for use in a 50Hz line frequency signal processing application.

In addition to the signal model, the filter type is also generalized. This yields design methods for finding the optimal parameters of differentiators, predictive differentiators and integrators, among others. The asymptotic noise gain of generalized predictors is derived and found to depend on the largest modulus of the complex numbers which specify the signal model. Furthermore, an efficient implementation for generalized FIR predictors is derived.

##### Kokoelmat

- Väitöskirjat [4136]