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FIXED-MEMORY POLYNOMIAL FILTER 5.1 INTRODUCTION In Section 1.2.10 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 (4.1-32). As in Section 1.2.10 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 (4.1-1a). 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 5.1 INTRODUCTION In Section 1.2.10 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 4.1-32 . As in Section 1.2.10 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 4.1-1a . The state vector is given by 4.1-2 . We first use a direct approach that involves representing x t by an arbitrary mth polynomial and applying 4.1-32 5 pp. 225-228 . This approach is given in Section 5.2. This direct approach unfortunately requires a matrix inversion. In Section 4.3 we developed the voltage-processing approach which did not require a matrix inversion. In Section 5.3 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 6.3 for the growing-memory filter. The approach of Section 5.3 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 14.4. In so doing better insight into the Legendre orthogonal polynomial fit approach will be obtained. 205 206 FIXED-MEMORY POLYNOMIAL FILTER 5.2 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