Correction: An Efficient Algorithm for Maneuvering Target Tracking (IEEE Signal Processing Magazine (2021) 38:1 (122-130) DOI: 10.1109/MSP.2020.3029386)

This article contributes comments and corrections to the technical content of "An Efficient Algorithm for Maneuvering Target Tracking,"by Arman Kheirati Roonizi [1]. In the "The Singer Acceleration Model"section of [1], Roonizi wrote: It is worth noting that as the maneuver time constant τ increases, the acceleration becomes a white noise jerk model, and the Singer model reduces to the CV model. In cases where the maneuver time constant decreases, the acceleration becomes white noise, and the Singer model reduces to the CA model. The correct version has the words constant velocity (CV) and constant acceleration (CA) interchanged in these sentences [2, Sec. 4(D)]. In addition, although the maneuver time constant is usually denoted with τ (and its reciprocal with a) in the literature, τ had already been used in the article to denote the argument of the autocorrelation function of the target acceleration. So, in the preceding sentence, τ should either be omitted or replaced with 1/α. The correct version can be written as: It is worth noting that as the maneuver time constant (1/α) increases, the acceleration becomes a white noise jerk model, and the Singer model reduces to the CA model. In cases where the maneuver time constant decreases, the acceleration becomes white noise, and the Singer model reduces to the CV model. In the same section, the author wrote, "The target acceleration is described by the following linear time-invariant model: ak+1 =-βak + wk, where wk is zero-mean white Gaussian noise with variance σ2 (1-β2), β= e-αTs and Ts is the sampling period."Under this parametrization, the Singer acceleration model is correctly written without the minus sign, as follows [2, Eq. 27]: ak+1 = βak + wk. The expression Ts ((1 - β),α2) which appears in the third row of (6a), in the definition of ζ3 following (7), and at the (first row, third column) position of the matrix given inside the solid rectangle on [1, p. 124], needs to be corrected as Ts ((1 - β)/α). A coefficient of two appears erroneously at the beginning of [1, Eq. (6a)], and a parenthesis is missing at the end of [1, Eq. (6a)]. For clarity, the correct form of (6a) is (equation presented) the correct form of ζ3 is (equation presented) and the correct form of the matrix equation in the box at the bottom of [1, p. 124] is given in (1). In [1, Eq. 9], the author wrote, "The optimal solution is [7] (equation presented) The matrix sum FTF +λγTγ that appears in the preceding may not be invertible. (formula presented) As seen from their definitions preceding [1, Eq. 9], F and γ are rectangular matrices with sizes (N - 2) × N and (N - 3) × N, respectively. Therefore, FTF can be of rank (N - 2) at most. Likewise, FTF can be of rank (N - 3) at most. At the beginning of the "A Causal Filter"section in [1], the author wrote, "The matrix M=(FTF+λγTγ)-1 FTF is a symmetric positive definite real matrix."As mentioned, F is an (N - 2) × N rectangular matrix with at least a 2D null space. This means there exists y 0 such that yTMy = 0. Hence, M is not positive definite. In addition, for a sufficiently small sampling time Ts, the author argues that FTF can be replaced by the identity matrix (see [1, Eq. 10]). However, a rigorous proof of this statement is not provided. As noted, FTF can be of rank (N - 2) at most. Hence, the claim about its approximation with the full-rank N × N identity matrix needs to be substantiated. The claims following [1, Eq. 9] are based on the assumption that the sampling time Ts is sufficiently small, characterized by the author with the condition Ts ≪ 1. However, it is customary in the literature to compare the sampling time Ts with the maneuver time constant 1/α instead of simply letting Ts go to arbitrarily small values. For example, in [3, Eqs. 2.9-9, 2.9-11, and 2.9-12], the state transition matrix and maneuver excitation covariance matrix are explicitly specified for aTs % 1/2 and aTs ≪ 1. Setting Ts too small leads to a trivial constant acceleration model according to a Newtonian matrix with vanishing noise. Finally, the trick employed in the article to convert the problem into an independent difference model is based on the z transform, which is a standard technique in the analysis of discrete-time linear dynamical systems. This connection is not explicitly mentioned in the article.