*2.3. G-MODEL*

A compact polarized GA model for the vector-sensor array was proposed in [21], named G-MODEL, which models the six-component outputs of a vector sensor holistically using a multi-vector in G3. Suppose there are *K* narrow-band, far-field and uncorrelated sources with wavelength *λ* impinging on an array, which includes *Q* vector sensors. Define *θk* ∈ [0, <sup>2</sup>*π*), *φk* ∈ [0, *<sup>π</sup>*), *γk* ∈ [0, *π*/2) and *ηk* ∈ [−*π*, *π*) are the azimuth angle, elevation angle, polarization amplitude angle and phase difference angle of the *kth* source, respectively.

Define *uk* = cos *θk* sin *φke***1** + sin *θk* sin *φke***2** + cos *φke***3** as the unit vector (see Figure 1) of the *kth* source when it impinges on the sensor at the origin. *vk*1 = − sin *θke***1** + cos *θke***2** and *vk*2 = cos *θk* cos *φke***1** + sin *θk* cos *φke***2** − sin *φke***3** are unit multi-vectors. The position vector of the *qth* sensor is *rq* = *rq*1*e***1** + *rq*2*e***2** + *rq*3*e***3**. The output of the *qth* vector sensor in the array is denoted by [21]

$$Y\_{EH}^{(q)}(t) = \sum\_{k=1}^{K} X\_q(\theta\_k, \phi\_k) V\_k P\_k S\_k(t) + N\_{EH}^{(q)}(t),\tag{7}$$

where *Xq*(*<sup>θ</sup>k*, *φk*) = *ee*<sup>123</sup> 2*πλ* (cos *θk* sin *φkrq*1+sin *θk* sin *φkrq*2+cos *φkrq*3) is the spatial phase factor of the *kth* source incident on the *qth* vector sensor.

$$\begin{aligned} V\_k &= (1 + \mathfrak{u}\_k)[\mathfrak{v}\_{k1'} - \mathfrak{v}\_{k2}], \\ P\_k &= \left[ \begin{array}{c} \cos \gamma\_k \\ \sin \gamma\_k e^{\mathfrak{e}\_{123}\eta\_k} \end{array} \right], \\ S\_k(t) &= |S\_k(t)| \exp[\mathfrak{e}\_{123}(2\pi f\_k t)]. \end{aligned}$$

In next section, the GA-ESPRIT algorithm is deduced based on the G-MODEL.

**Figure 1.** Direction vector of incident source.
