Electronically Scanned Arrays MATLAB® Modeling and Simulation

Appendix C: Triangular Grating Lobes Derivation

In Chapter 2, the grating lobe expression for a triangular grid of elements was shown to be (Skolnik 1990; Corey 1985):

$\begin{array}{l}\begin{array}{cc}{u}_m=u_o+m\frac{\text{λ}}{2d_x},& v_n=v_o+n\frac{\text{λ}}{2d_y}\end{array}\\ m,n=0,\pm 1,\pm 2,\dots \\ m+n\text{is}\text{even}\end{array}$

(C.1)

This expression is derived in Corey (1985); however, a more intuitive derivation will be shown here. We begin by considering a rectangular array with element spacing 2d_x and 2d_y, as shown in Figure C.1. In Chapter 2 it was shown that the grating lobes for this array occur at

$\begin{array}{ll}{u}_m=u_o+m\frac{\text{λ}}{(2d_x)},\hfill & m=0,\pm 1,\pm 2,\dots \hfill \\ u_n=u_o+n\frac{\text{λ}}{(2d_y)},\hfill & n=0,\pm 1,\pm 2,\dots \hfill \end{array}$

(C.2)

Next we consider another array of elements with the same 2d_x and 2d_y spacing, but offset by d_x and d_y, as shown in Figure C.2. This offset can be expressed mathematically with a complex ...