Polynomial Predictive Filters: Implementation and Applications

In this thesis, smoothness of sampled real-world signals is exploited through the application of polynomial predictive filters. The principal reason for employing the polynomial signal model is principally twofold: firstly, assuming that the sampling rate is adequate, all realworld signals exhibit piecewise polynomial-like behavior, and secondly, polynomial-based signal processing is computationally efficient. By definition, polynomial predictive filters provide estimates of future values of polynomial-like signals. Thus, the potential applications of this research include a vast number of different delay sensitive operations on measurements like temperature, position, velocity, or power, especially in control engineering field. The polynomial-based predictive signal processing is a well-known technique, but polynomial-predictive filters have had severe drawbacks, which have hindered their application; their white noise attenuation is generally low, or they exhibit considerable passband gain peaks, rendering them unattractive for most applications. It has been possible to design IIR polynomial predictors, which exhibit applicable magnitude response properties, but the severe problem with them, as well as with the FIR polynomial predictors, has been that they have generally not been implementable in low-precision fixed-point environments because of their coefficient quantization sensitivity. In this thesis, coefficient quantization error-free designs of both FIR and IIR polynomial predictors are presented, thus providing methods for overcoming the above drawbacks and design problems. Polynomial differentiators are closely related to polynomial predictors; they are derived in a similar fashion, have design problems of a similar nature, and have applications in the control field. Both of these two filter types are discussed in this thesis; the proposed design methods are applicable to both of them. The implementation aspects of polynomial predictors and differentiators investigated here are also connected to the practical requirements of the application, namely delay alleviation in closed loop transmitter power control of multiuser mobile communications systems. Particularly, if predictive received power level estimation is implemented in handheld mobile terminals, this application specifies the implementation criteria as requirements on low imposed computational burden, low power consumption, and compact hardware size. All these criteria are met by providing the desired functionality using a small number of fixedpoint arithmetic operations. Taking into account the results presented in this thesis, polynomial prediction fulfills these criteria. In this thesis, digital filter design methodologies are advanced by first-time introduction of exact low-degree polynomial prediction and discrete time differentiation in low-precision fixed-point computing environments, with, for example, 8 or 16 bits. Polynomial prediction is shown advantageous in the closed loop transmitter power control system application, and in comparisons with more complex and flexible predictors, it is shown to be a highly efficient method for this particular application. This thesis is seen as contributing to advances in practical polynomial predictor and differentiator design methods, and thereafter studies the application of polynomial predictors in mobile communications system transmitter power control. This research will be of interest to signal processing, control, and communications engineers and researchers alike. ii Polynomial-Predictive Filters: Implementation and Applications