Optimal Design of Digital IIR and FIR Filters Using Complex Flatness Constraints: A Unified Approach

Fixed-Gain Augmented-State Tracking-Filters

“…The use of this observer results in a SISO filter with a "flatness" order of 𝐿 dc , 𝐿 wb and 𝐿 pi , at the critical frequencies 𝜔 𝑐 = 0 (dc), 𝜔 𝑐 = ±Ω𝑇 𝑠 (wideband) and 𝜔 𝑐 = 𝜋 (Nyquist) respectively [2], [32], where 𝐿 dc = 𝐾 tgt , 𝐿 wb = 𝐾 man and 𝐿 pi = 𝐾 int , for the complex frequency response 𝐻(𝜔) = ℋ(𝑧)| 𝑧=𝑒 𝑖𝜔 , such that where 𝜌 𝑙 (𝜔 𝑐 ) = 0 for 𝜔 𝑐 = 𝜋 (stopband). ( 9)…”

“…For example, a fixed-lag smoother, with 𝐷 = 0 and a lag of 𝑞 samples, configured using 𝐾 tgt = 𝐾 int = 𝐿 and 𝐾 man = 0, arXiv:1709.02542 [cs.SY] results in 𝜌 𝑙 (0) = (−𝑖𝑞) 𝑙 and 𝜌 𝑙 (𝜋) = 0, for 0 ≤ 𝑙 < 𝐿. Flatness ensures that the filter behaves exactly/approximately as required at/near the critical design frequencies [2], [32]. Satisfaction of the dc constraints in (9) guarantees unbiased estimation of the kinematic target states, at steady state, in the absence of target maneuvers.…”

“…The observers and filters described in this paper may be suitable for point-target tracking and low-level imageprocessing functions (respectively) within a computer-vision system, for video moving-target indication, on an airborne platform [32], [35], [36]. Fixed-gain trackers are suitable in these systems, for the following reasons: High data-rates (e.g.…”

“…accurate kinematic state estimation, high-frequency noise attenuation and circular manoeuvre trackingby the appropriate placement of filter zeros. In this paper, they are placed indirectly through the selection of physical process models, which maintains a connection with the kinematic states; whereas a linear combination of analytical basis functions is used in [2] & [32].…”