FIXED-MEMORY POLYNOMIAL FILTER INTRODUCTION In Section we presented the growing-memory g–h filter. For n fixed this filter becomes a fixed-memory filter with the n most recent samples of data being processed by the filter, sliding-window fashion. In this chapter we derive a higher order form of this filter. We develop this higher order fixed-memory polynomial filter by applying the least-squares results given by (). As in Section we assume that only measurements of the target range, designated as xðtÞ, are available, that is, the measurements are onedimensional, hence r ¼ 0 in (). The state vector is given. | Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright 1998 John Wiley Sons Inc. ISBNs 0-471-18407-1 Hardback 0-471-22419-7 Electronic 5 FIXED-MEMORY POLYNOMIAL FILTER INTRODUCTION In Section we presented the growing-memory g-h filter. For n fixed this filter becomes a fixed-memory filter with the n most recent samples of data being processed by the filter sliding-window fashion. In this chapter we derive a higher order form of this filter. We develop this higher order fixed-memory polynomial filter by applying the least-squares results given by . As in Section we assume that only measurements of the target range designated as x t are available that is the measurements are onedimensional hence r 0 in . The state vector is given by . We first use a direct approach that involves representing x t by an arbitrary mth polynomial and applying 5 pp. 225-228 . This approach is given in Section . This direct approach unfortunately requires a matrix inversion. In Section we developed the voltage-processing approach which did not require a matrix inversion. In Section we present another approach that does not require a matrix inversion. This approach also has the advantage of leading to the development of a recursive form to be given in Section for the growing-memory filter. The approach of Section involves using the discrete-time orthogonal Legendre polynomial DOLP representation for the polynomial fit. As indicated previously the approach using the Legendre orthogonal polynomial representation is equivalent to the voltage-processing approach. We shall prove this equivalence in Section . In so doing better insight into the Legendre orthogonal polynomial fit approach will be obtained. 205 206 FIXED-MEMORY POLYNOMIAL FILTER DIRECT APPROACH USING NONORTHOGONAL mTH-DEGREE POLYNOMIAL FIT Assume a sequence of L 1 one-dimensional measurements given by Y yn yn-1 . yn-L T 5-2-1 with n being the last time
