2.1.2. Sensor Observation Model

The radar can provide range and bearing measurements of the target's centroid, and the observation model of kinematic (position) measurement can be described by

$$z\_k = \begin{bmatrix} r\_k \\ \beta\_k \end{bmatrix} = \begin{bmatrix} \sqrt{\mathbf{x}\_k^2 + \mathbf{y}\_k^2} + v\_{1,k} \\ \tan^{-1}(y\_k/\mathbf{x}\_k) + v\_{2,k} \end{bmatrix} = h(\mathbf{x}\_k) + \mathbf{v}\_k \tag{3}$$

where *h*(·) is the kinematic observation function and *rk* and β*<sup>k</sup>* represent noisy measurements of target range and bearing at time *<sup>k</sup>*, respectively. *<sup>v</sup><sup>k</sup>* <sup>=</sup> *v*1,*<sup>k</sup> v*2,*<sup>k</sup>* <sup>T</sup> denotes the corresponding zero-mean observation noise with covariance matrix *R* = *E*[*vkv*<sup>T</sup> *<sup>k</sup>* ] = *diag*[σ<sup>2</sup> *<sup>r</sup>*, σ<sup>2</sup> β].

As shown in Figure 1, for the low-speed maritime target, the heading is almost aligned with the axial direction of the target body because of its limited maneuverability. Under this condition, the aspect angle φ*<sup>k</sup>* of the target can be obtained as

$$
\phi\_k = \theta\_k - \beta\_k \tag{4}
$$

where θ*<sup>k</sup>* = tan−1( . *yk*/ . *xk*) is the heading angle.

**Figure 1.** Illustration of target aspect angle.

An equivalent expression of aspect angle is in the form of

$$\phi\_k = \cos^{-1}\left(\frac{\langle pos\_k, vel\_k \rangle}{||pos\_k|| \cdot ||\text{vel}\_k||}\right) = \cos^{-1}\left(\frac{x\_k \dot{x}\_k + y\_k \dot{y}\_k}{\sqrt{x\_k^2 + y\_k^2}\sqrt{\dot{x}\_k^2 + \dot{y}\_k^2}}\right) \tag{5}$$

The 3D-SCM is an equivalent of the target in geometry space to the radar response in the electromagnetic field. It provides a concise and physically relevant description of the target's scattering through a set of representative scattering parts, and thus a more effective way to characterize the target's electromagnetic scattering behavior. The 3D-SCM consists of a set of scattering center with a specific position, amplitude and type parameters, and it can be represented by

$$\mathbf{S} = \{a\_{n\prime} \alpha\_{n\prime} \ge\_{n\prime} y\_{n\prime} z\_n\}\_{n=1}^{N} \tag{6}$$

where *an* is the amplitude of *n*th scattering center, α*<sup>n</sup>* is a frequency-dependent factor, (*xn*, *yn*, *zn*) is the corresponding 3D spatial position in the target body coordinates and *N* is the number of scatters involved in the model. For a specific target class *c*, the associated 3D-SCM can be denoted as *Sc*.

The whole target's backscattering with respect to radar instantaneous frequency *f*, viewing angles (i.e., azimuth angle φ and elevation angle γ) can be expressed as [13]

$$E(f\_{\boldsymbol{\gamma}}\boldsymbol{\phi}, \boldsymbol{\gamma}, \mathbf{S}) = \sum\_{n=1}^{N} \left( \mathbf{j}f / f\_{\boldsymbol{\varepsilon}} \right)^{a\_n} a\_n (\boldsymbol{\phi}, \boldsymbol{\gamma}) \cdot \exp \left( -\mathbf{j}4\pi (\mathbf{x}\_n \cos \boldsymbol{\gamma} \cos \boldsymbol{\phi} + y\_n \cos \boldsymbol{\gamma} \sin \boldsymbol{\phi} + z\_n \sin \boldsymbol{\gamma}) / \lambda \right) \tag{7}$$

where <sup>λ</sup> is the wavelength, *fc* is the central frequency of signal, <sup>j</sup> <sup>=</sup> <sup>√</sup> −1 is the imaginary unit and *an*(φ, γ) represents the amplitude of *n*th scattering center which may change with φ and γ.

At a specific viewing angle of (φ, γ), the corresponding projection position of the *n*th scattering center at the down-range direction is

$$\tau\_n(\phi, \gamma) = x\_n \cos \gamma \cos \phi + y\_n \cos \gamma \sin \phi + z\_n \sin \gamma \tag{8}$$

Assuming that the bandwidth of the radar signal is *B* and the *i*th discrete frequency point is

$$f\_i = f\_\varepsilon - B / 2 + i \cdot \Delta F, \quad i = 0, 1, \cdots, I \tag{9}$$

where Δ*F* is the frequency interval and *I* + 1 is the total number of frequency points.

Then, the frequency response of the *i*th frequency point can be written as *Ei* = *E*(*fi*,φ, γ, *S*). After a direct operation of inverse discrete Fourier transform (IDFT) on frequency response sequence *<sup>E</sup>* = [*E*0, *<sup>E</sup>*1, ··· , *EI*], the desired HRRP can be immediately obtained.

When the target's motion is restricted on the 2D plane, the observation model of HRRP is represented as

$$\begin{array}{lcl} d &= \mathcal{g}(\phi, \mathbf{S}) + \mathfrak{n} \\ &= IDFT[E\_i = E(f\_i, \phi\_\prime \mathcal{y}\_\prime, \mathbf{S}), i = 0, 1, \cdots, I] + \mathfrak{n} \\ &= IDFT(E) + \mathfrak{n} \end{array} \tag{10}$$

where *d* denotes the (*I*+1)-dimensional measurement of the HRRP and each component of *d* corresponds to one range resolution cell, *g*(φ, *S*) - *IDFT*(*E*) denotes the compact form of observation function, *fi* and *<sup>S</sup>* are known parameters, <sup>γ</sup> <sup>≈</sup> 0, <sup>φ</sup> can be obtained through Equations (4) or (5) and *<sup>n</sup>* is the observation noise vector.
