Smoothing Irregular data using Polynomial Filters Pieter Reyneke * , Norman Morrison † , Derrick Kourie ‡ and Corn´ e de Ridder § * Department Radar and Imaging Systems, Denel Dynamics, Irene, 0157, South Africa † Department of Electrical Engineering, University of Cape Town, Cape Town, 7700, South Africa ‡ Fastar Research Group, University of Pretoria, Pretoria, 0002, South Africa § School of Computing, University of South Africa, Pretoria, 0002, South Africa Email: pieter.reyneke@gmail.com Abstract—To make provision for irregularly occurring updates a “variable-step” polynomial filter is derived which improves the smoothing-, and coasting capabilities of the 1-step predictor and the current-estimate polynomial filters. Similar to the original filters, the variable-step polynomial filter comprises an auto- initialising expanding memory filter which later switches to a fading memory filter. Results are compared by running the proposed and original filter versions in parallel. Keywords—Radar tracking filters, Polynomial approximation, Smoothing, Extrapolation, State estimation I. I NTRODUCTION The 1-step predictor and current-estimate filters introduced by Morrison in [1] and [2] both recursively calculate a least squares approximation of a process model, determining the state of a polynomial curve fitted over noisy observations. In the present context, recursive refers to adding one observation at a time. These filters are subject to the constraint that observations should be evenly spaced. However, in natural environments, measurements cannot always be assigned to an integer type time batch without losing accuracy. The reasons for this are threefold: • High-fidelity time stamps are usually measured during real-time process control as floating point values. • Floating point time values more accurately position up- dates in time, leading to less ambiguity caused by natural variations (related, for example, to temperature, target clutter, occlusion, etc.) • Detector or algorithm design may result in non- deterministic jumps in the update intervals, or in uneven even/odd time-interval symmetry. Examples where vari- able update intervals can be expected include: nodding algorithms, zig-zag sweep detectors, linear sensor inte- gration and asynchronous mode changing. For above mentioned reasons this research proposes a variable- step extension to the polynomial filters derived in [1] and [2]. A. Background — Polynomial filters Recursive polynomial filters calculate either a least squares solution (the expanding memory polynomial (EMP)) or a weighted least squares solution with the weight (θ ∈ (0, 1)) fading the previous state estimate, Z (t)= θZ (t -δ)+Γ(1 -θ) (the fading memory polynomial (FMP)). For the EMP, Γ(n) (see Section IV-A), based on orthonormal discrete Legendre polynomials, is used to update a least squares fit. In the case of FMP, Γ(θ) (see Section IV-B), based on orthonormal discrete Laguerre polynomials, is used to update weighted least squares update fit with an update weight of (1 - θ). This is done to realise a recursive autoregressive state update in each case, as derived in [2] — i.e. one sample at a time is added to the current state vector. Morrison [1], distinguishes between a current-estimate and a 1-step predictor. Equations 1 and 2 show the computation of the predicted state, Z * n and the error term e n for both the current-estimate filter and the 1-step predictor filter. Z * n = Φ(1)Z n-1 , ... (predict state Z * n ) (1) e n = y n - z * 0 n , ... (calculate error term e n ) (2) However, the formula for updating differs for the two respec- tive cases, as shown in equations 3 and 4. Furthermore, in each case below, either Γ i (n) or Γ i (θ) needs to be used in the place of Γ i to do the update for the EMP or FMP respectively, see Sections IV-A and IV-B. The current-estimate filter: Use either Γ i (n) or Γ i (θ) and do the update: Z n = Z * n +Γ i e n (3) The 1-step predictor filter: Use either Γ i (n) or Γ i (θ) and do the update: Φ(-1)Z n+1,n = Z * n +Γ i e n (4) This last step, Equation 4, written out for the 2 nd order, 1-step predictor update, renders: z * 2 n+1,n = z * 2 n,n-1 +Γ 2 e n z * 1 n+1,n = z * 1 n,n-1 +2z * 2 n+1,n +Γ 1 e n z * 0 n+1,n = z * 0 n,n-1 + z * 1 n+1,n - z * 2 n+1,n +Γ 0 e n The remainder of this paper is laid out as follows. Section II provides the classical linear tracking differential equation. In Section III the motivation for a change in state transition matrix is presented. A formal explanation of the extension of the current-estimate filter to a variable-step polynomial filter can be found in Section IV. Section V reports on results obtained from trial runs on simulated polynomial data thereby verifying the prediction capability of a variable-step implementation. Section VI presents the smoothing results of very noisy and irregular, real data. Finally, results obtained during missile testing are provided in VII, before concluding the paper in Section VIII.