**3.3 Array factors of some well-known sparse arrays**

According to the principle of pattern multiplication, the overall array response is the product of the array factor and the element pattern. The element pattern for an isotropic antenna is unity. Hence, in an array of isotropic antenna elements, the farfield pattern depends only on the array factor. The array factor of sparse arrays is conveniently evaluated using the element positions *xn* as given below

$$AF(\phi) = \sum\_{n \in \text{sparse}} \mathcal{e}^{ik\chi\_n \cos \phi},\tag{6}$$

where, *xn* denotes the actual grid-point location of the *n*th element in the sparse array. *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> <sup>λ</sup>* denotes the wavenumber and *ϕ* denotes the azimuth angle. For an MRA with sensors at {0, 1, 4, 6}, the values of *xn* in the above formula would be {0, *d*, 4*d*, 6*d*}, respectively.

To get further idea on the characteristics of sparse arrays, the array factors of a few prominent sparse arrays given in **Table 1** were evaluated using Eq. (6) and plotted using MATLAB. **Table 3** lists the sparse arrays considered and their 3-dB beamwidths. **Figure 3** shows the plot of array factors of the considered sparse arrays. It can be observed that MRAs provide the narrowest main beam


**Table 3.**

*Beamwidths for the 10-element sparse arrays listed in Table 1.*

**Figure 3.** *Array factors of sparse arrays listed in Table 3.*

characteristics for a given number of sensors as they possess the largest aperture among all the sparse arrays considered here. It has to be noted that the MHA can offer even finer beamwidth than MRAs.

It would be good to have a few more figures representing the weight functions of the above sparse arrays and their DOA estimation performance when used with co-array MUSIC. However, they are omitted from this chapter for lack of space.
